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Journal articles on the topic 'Finite index subgroups'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Srinivasan, S. "Maximal subgroups of finite groups." International Journal of Mathematics and Mathematical Sciences 13, no. 2 (1990): 311–14. http://dx.doi.org/10.1155/s016117129000045x.

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In finite groups maximal subgroups play a very important role. Results in the literature show that if the maximal subgroup has a very small index in the whole group then it influences the structure of the group itself. In this paper we study the case when the index of the maximal subgroups of the groups have a special type of relation with the Fitting subgroup of the group.

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2

Kappe, Luise-Charlotte. "Finite coverings by 2-engel groups." Bulletin of the Australian Mathematical Society 38, no. 1 (August 1988): 141–50. http://dx.doi.org/10.1017/s0004972700027350.

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Bear's characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups if the starting point for this invertigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2-Engel subgroups are exactly the 3-Engel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of thes variety in the group has finite index.

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3

De Falco, M., F. De Giovanni, C. Musella, and Y. P. Sysak. "The structure of groups whose subgroups are permutable-by-finite." Journal of the Australian Mathematical Society 81, no. 1 (August 2006): 35–48. http://dx.doi.org/10.1017/s1446788700014622.

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AbstractA subgroup H of a group G is said to be permutable if HX = XH for each subgroup X of G, and the group G is called quasihamiltonian if all its subgroups are permutable. We shall say that G is a Q F-group if every subgroup H of G contains a subgroup K of finite index which is permutable in G. It is proved that every locally finite Q F-group contains a quasihamiltonian subgroup of finite index. In the proof of this result we use a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups: if G is a locally finite group such that H/HG is finite for every subgroup H, then G contains an abelian subgroup of finite index.

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4

DE GIOVANNI, F., M. MARTUSCIELLO, and C. RAINONE. "LOCALLY FINITE GROUPS WHOSE SUBGROUPS HAVE FINITE NORMAL OSCILLATION." Bulletin of the Australian Mathematical Society 89, no. 3 (November 20, 2013): 479–87. http://dx.doi.org/10.1017/s000497271300097x.

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AbstractIf $X$ is a subgroup of a group $G$, the cardinal number $\min \{ \vert X: X_{G}\vert , \vert {X}^{G} : X\vert \} $ is called the normal oscillation of $X$ in $G$. It is proved that if all subgroups of a locally finite group $G$ have finite normal oscillation, then $G$ contains a nilpotent subgroup of finite index.

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5

Nikolov, Nikolay, and Dan Segal. "Finite index subgroups in profinite groups." Comptes Rendus Mathematique 337, no. 5 (September 2003): 303–8. http://dx.doi.org/10.1016/s1631-073x(03)00349-2.

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6

M�ller, Thomas. "Counting free subgroups of finite index." Archiv der Mathematik 59, no. 6 (December 1992): 525–33. http://dx.doi.org/10.1007/bf01194843.

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7

Haglund, Frédéric. "Finite index subgroups of graph products." Geometriae Dedicata 135, no. 1 (May 27, 2008): 167–209. http://dx.doi.org/10.1007/s10711-008-9270-0.

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8

NIKOLAEV, ANDREY V., and DENIS E. SERBIN. "FINITE INDEX SUBGROUPS OF FULLY RESIDUALLY FREE GROUPS." International Journal of Algebra and Computation 21, no. 04 (June 2011): 651–73. http://dx.doi.org/10.1142/s0218196711006388.

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Using graph-theoretic techniques for f.g. subgroups of Fℤ[t] we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. As an application we obtain an analogue of Greenberg–Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. nonabelian subgroup of a f.g. fully residually free group is of finite index in its normalizer and commensurator.

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9

Szepietowski, Błażej. "On finite index subgroups of the mapping class group of a nonorientable surface." Glasnik Matematicki 49, no. 2 (December 18, 2014): 337–50. http://dx.doi.org/10.3336/gm.49.2.08.

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10

HARTUNG, RENÉ. "COSET ENUMERATION FOR CERTAIN INFINITELY PRESENTED GROUPS." International Journal of Algebra and Computation 21, no. 08 (December 2011): 1369–80. http://dx.doi.org/10.1142/s0218196711006637.

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We describe an algorithm that computes the index of a finitely generated subgroup in a finitely L-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely L-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group, and the Hanoi 3-group.

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11

Monakhov, Victor S., and Alexander A. Trofimuk. "Finite groups with two supersoluble subgroups." Journal of Group Theory 22, no. 2 (March 1, 2019): 297–312. http://dx.doi.org/10.1515/jgth-2018-0150.

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AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

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12

Shusterman, Mark. "Groups with positive rank gradient and their actions." Mathematica Slovaca 68, no. 2 (April 25, 2018): 353–60. http://dx.doi.org/10.1515/ms-2017-0106.

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Abstract We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A, B ≤ G of infinite index, one can find a finite index subgroup B0 of B such that [G : 〈A ∪ B0〉] = ∞. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover G. We construct a transitive virtually faithful action of G such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite groups with positive rank gradient.

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13

BALLESTER-BOLINCHES, A., J. C. BEIDLEMAN, and R. ESTEBAN-ROMERO. "PRIMITIVE SUBGROUPS AND PST-GROUPS." Bulletin of the Australian Mathematical Society 89, no. 3 (July 18, 2013): 373–78. http://dx.doi.org/10.1017/s0004972713000592.

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AbstractAll groups considered in this paper are finite. A subgroup $H$ of a group $G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of $G$ containing $H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of $G$ has index a power of a prime if and only if $G/ \Phi (G)$ is a solvable PST-group. Let $\mathfrak{X}$ denote the class of groups $G$ all of whose primitive subgroups have prime power index. It is established here that a group $G$ is a solvable PST-group if and only if every subgroup of $G$ is an $\mathfrak{X}$-group.

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14

HADAD, UZY. "ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)." International Journal of Algebra and Computation 22, no. 03 (May 2012): 1250026. http://dx.doi.org/10.1142/s0218196712500269.

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We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

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15

Lubotzky, Alexander. "On Finite Index Subgroups of Linear Groups." Bulletin of the London Mathematical Society 19, no. 4 (July 1987): 325–28. http://dx.doi.org/10.1112/blms/19.4.325.

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16

Berrick, A. J., V. Gebhardt, and L. Paris. "Finite index subgroups of mapping class groups." Proceedings of the London Mathematical Society 108, no. 3 (August 5, 2013): 575–99. http://dx.doi.org/10.1112/plms/pdt022.

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17

Rosenmann, Amnon. "SUBGROUPS OF FINITE INDEX AND THEfc-LOCALIZATION." Communications in Algebra 29, no. 5 (April 30, 2001): 1983–91. http://dx.doi.org/10.1081/agb-100002162.

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18

Grunewald, F. J., D. Segal, and G. C. Smith. "Subgroups of finite index in nilpotent groups." Inventiones Mathematicae 93, no. 1 (February 1988): 185–223. http://dx.doi.org/10.1007/bf01393692.

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19

Kapovich, Ilya, and Hamish Short. "Greenberg's Theorem for Quasiconvex Subgroups of Word Hyperbolic Groups." Canadian Journal of Mathematics 48, no. 6 (December 1, 1996): 1224–44. http://dx.doi.org/10.4153/cjm-1996-065-6.

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AbstractAnalogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

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20

Azarian, Mohammad K. "Near Frattini subgroups of residually finite generalized free products of groups." International Journal of Mathematics and Mathematical Sciences 26, no. 2 (2001): 117–21. http://dx.doi.org/10.1155/s0161171201005397.

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LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

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21

Lubotzky, Alexander, and Avinoam Mann. "Residually finite groups of finite rank." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 3 (November 1989): 385–88. http://dx.doi.org/10.1017/s0305004100068110.

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The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:Theorem 1. A residually finite group of finite rank is virtually locally soluble.

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22

Kurdachenko, Leonid A., and Igor Ya Subbotin. "THE GROUPS WHOSE SUBGROUPS ARE ALMOST ASCENDANT." Asian-European Journal of Mathematics 06, no. 01 (March 2013): 1350014. http://dx.doi.org/10.1142/s1793557113500149.

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A subgroup H of a group G is called almost ascendant if H is ascendant in a subgroup K having finite index in G. We describe the structure of periodic groups whose subgroups are almost ascendant. The main result of this paper is following theorem. Let G be a periodic group whose subgroups are almost ascendant. Then G contains a normal finite subgroup K such that every subgroup of G/K is ascendant in G/K. In particular, G/K satisfies the normalizer condition.

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23

Brodie, Marc A., and Luise-Charlotte Kappe. "Finite coverings by subgroups with a given property." Glasgow Mathematical Journal 35, no. 2 (May 1993): 179–88. http://dx.doi.org/10.1017/s0017089500009733.

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Let be a group-theoretic property. We say a group has a finite covering by -subgroups if it is the set-theoretic union of finitely many -subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups, n-abelian subgroups or 2-central subgroups.R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC (G)] finite implies the existence of a finite covering by subgroups of nilpotency class c, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, but Z2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.

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24

MENG, HANGYANG, and XIUYUN GUO. "OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS." Bulletin of the Australian Mathematical Society 99, no. 1 (August 30, 2018): 83–88. http://dx.doi.org/10.1017/s0004972718000904.

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A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

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25

KUTNAR, KLAVDIJA, DRAGAN MARUŠIČ, JIANGTAO SHI, and CUI ZHANG. "FINITE GROUPS WITH SOME NON-CYCLIC SUBGROUPS HAVING SMALL INDICES IN THEIR NORMALIZERS." Journal of Algebra and Its Applications 13, no. 04 (January 9, 2014): 1350141. http://dx.doi.org/10.1142/s0219498813501417.

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In this paper, it is shown that a finite group G is always supersolvable if |NG(H) : H| ≤ 2 for every non-cyclic subgroup H of G of prime-power order. Also, finite groups with all supersolvable non-cyclic subgroups being self-normalizing, and finite p-groups with all non-cyclic proper subgroups being of prime index in their normalizers are completely classified.

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26

Rozov, A. V. "On the Residual Finiteness of Some Generalized Products of Soluble Groups of Finite Rank." Modeling and Analysis of Information Systems 20, no. 1 (March 18, 2015): 124–32. http://dx.doi.org/10.18255/1818-1015-2013-1-124-132.

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Let G be a free product of residually finite virtually soluble groups A and B of finite rank with an amalgamated subgroup H, H 6= A and H 6= B. And let H contains a subgroup W of finite index which is normal in both A and B. We prove that the group G is residually finite if and only if the subgroup H is finitely separable in A and B. Also we prove that if all subgroups of A and B are finitely separable in A and B, respectively, all finitely generated subgroups of G are finitely separable in G.

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27

Shalev, Aner. "Groups Whose Subgroup Growth is Less than Linear." International Journal of Algebra and Computation 07, no. 01 (February 1997): 77–91. http://dx.doi.org/10.1142/s0218196797000071.

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Let G be a residually finite group and let an(G) denote the number of index n subgroups of G. It is shown that an(G)/n →0 if and only if G has a finite index central subgroup whose finite quotients are all cyclic. As an application we show that the degree of a group of polynomial subgroup growth cannot lie strictly between 0 and 1.

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28

Reid, Colin D. "Distal actions on coset spaces in totally disconnected locally compact groups." Journal of Topology and Analysis 12, no. 02 (September 28, 2018): 491–532. http://dx.doi.org/10.1142/s1793525319500523.

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Let [Formula: see text] be a totally disconnected locally compact (t.d.l.c.) group and let [Formula: see text] be an equicontinuously (for example, compactly) generated group of automorphisms of [Formula: see text]. We show that every distal action of [Formula: see text] on a coset space of [Formula: see text] is a SIN action, with the small invariant neighborhoods arising from open [Formula: see text]-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups of a t.d.l.c. group. For example, it follows that for every compactly generated subgroup [Formula: see text] of [Formula: see text], there is a compactly generated open subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and such that every open subgroup of [Formula: see text] containing a finite index subgroup of [Formula: see text] contains a finite index subgroup of [Formula: see text]. We also show that for a large class of closed subgroups [Formula: see text] of [Formula: see text] (including for instance all closed subgroups [Formula: see text] such that [Formula: see text] is an intersection of subnormal subgroups of open subgroups), every compactly generated open subgroup of [Formula: see text] can be realized as [Formula: see text] for an open subgroup of [Formula: see text].

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29

Cornulier, Yves. "Subgroups approximatively of finite index and wreath products." Groups, Geometry, and Dynamics 8, no. 3 (2014): 775–88. http://dx.doi.org/10.4171/ggd/247.

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30

Dudkin, F. A. "Subgroups of finite index in Baumslag–Solitar groups." Algebra and Logic 49, no. 3 (July 2010): 221–32. http://dx.doi.org/10.1007/s10469-010-9091-8.

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31

Hosaka, Tetsuya. "Parabolic subgroups of finite index in Coxeter groups." Journal of Pure and Applied Algebra 169, no. 2-3 (April 2002): 215–27. http://dx.doi.org/10.1016/s0022-4049(01)00068-8.

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32

Clemens, Miles A., Branton J. Campbell, Stephen P. Humphries, and H. T. Stokes. "Finite-index normal subgroups of crystallographic space groups." Acta Crystallographica Section A Foundations and Advances 73, a1 (May 26, 2017): a373. http://dx.doi.org/10.1107/s0108767317096362.

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33

Yin, Xia, and Xianhua Li. "THE NORMAL INDEX OF SUBGROUPS IN FINITE GROUPS." Asian-European Journal of Mathematics 03, no. 03 (September 2010): 511–19. http://dx.doi.org/10.1142/s1793557110000325.

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34

Bergelson, Vitaly, and Daniel B. Shapiro. "Multiplicative subgroups of finite index in a ring." Proceedings of the American Mathematical Society 116, no. 4 (April 1, 1992): 885. http://dx.doi.org/10.1090/s0002-9939-1992-1095220-5.

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35

Wehrfritz, B. A. F. "Subgroups of prescribed finite index in linear groups." Israel Journal of Mathematics 58, no. 1 (February 1987): 125–28. http://dx.doi.org/10.1007/bf02764674.

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36

Garrido, Alejandra, and John S. Wilson. "On subgroups of finite index in branch groups." Journal of Algebra 397 (January 2014): 32–38. http://dx.doi.org/10.1016/j.jalgebra.2013.08.025.

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37

Skresanov, Saveliy V. "Subgroups of minimal index in polynomial time." Journal of Algebra and Its Applications 19, no. 01 (January 29, 2019): 2050010. http://dx.doi.org/10.1142/s0219498820500103.

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By applying an old result of Y. Berkovich, we provide a polynomial-time algorithm for computing the minimal possible index of a proper subgroup of a finite permutation group [Formula: see text]. Moreover, we find that subgroup explicitly and within the same time if [Formula: see text] is given by a Cayley table. As a corollary, we get an algorithm for testing whether or not a finite permutation group acts on a tree non-trivially.

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38

Pride, S. J., and Jing Wang. "Subgroups of finite index in groups with finite complete rewriting systems." Proceedings of the Edinburgh Mathematical Society 43, no. 1 (February 2000): 177–83. http://dx.doi.org/10.1017/s0013091500020794.

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AbstractWe show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).

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39

ABÉRT, MIKLÓS, and GÁBOR ELEK. "Dynamical properties of profinite actions." Ergodic Theory and Dynamical Systems 32, no. 6 (November 16, 2011): 1805–35. http://dx.doi.org/10.1017/s0143385711000654.

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AbstractWe study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

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40

Kryazheva, A. A. "Residual separability of subgroups in free products with amalgamated subgroup of finite index." Sibirskii matematicheskii zhurnal 60, no. 2 (July 19, 2018): 411–18. http://dx.doi.org/10.33048/smzh.2019.60.212.

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41

Kryazheva, A. A. "Residual Separability of Subgroups in Free Products with Amalgamated Subgroup of Finite Index." Siberian Mathematical Journal 60, no. 2 (March 2019): 319–24. http://dx.doi.org/10.1134/s0037446619020125.

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42

SNOPCE, ILIR, and PAVEL A. ZALESSKII. "Subgroup properties of Demushkin groups." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 1 (October 5, 2015): 1–9. http://dx.doi.org/10.1017/s0305004115000481.

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AbstractWe prove that a non-solvable Demushkin group satisfies the Greenberg–Stallings property, i.e., if H and K are finitely generated subgroups of a non-solvable Demushkin group G with the property that H ∩ K has finite index in both H and K, then H ∩ K has finite index in 〈H, K〉. Moreover, we prove that every finitely generated subgroup H of G has a ‘root’, that is a subgroup K of G that contains H with |K : H| finite and which contains every subgroup U of G that contains H with |U : H| finite. This allows us to show that every non-trivial finitely generated subgroup of a non-solvable Demushkin group has finite index in its commensurator.

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43

BAUMEISTER, BARBARA, and GIL KAPLAN. "c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS." Bulletin of the Australian Mathematical Society 86, no. 2 (April 3, 2012): 291–302. http://dx.doi.org/10.1017/s0004972712000081.

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Abstractc-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].

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44

BASSINO, FRÉDÉRIQUE, CYRIL NICAUD, and PASCAL WEIL. "RANDOM GENERATION OF FINITELY GENERATED SUBGROUPS OF A FREE GROUP." International Journal of Algebra and Computation 18, no. 02 (March 2008): 375–405. http://dx.doi.org/10.1142/s0218196708004482.

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We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity [Formula: see text] in the RAM model and [Formula: see text] in the bitcost model.

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45

Kondrat’ev, Anatoly S., Natalia V. Maslova, and Danila O. Revin. "Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal." Journal of Group Theory 23, no. 6 (November 1, 2020): 999–1016. http://dx.doi.org/10.1515/jgth-2020-0072.

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AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

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Hamilton, Thomas, and David Loeffler. "Congruence testing for odd subgroups of the modular group." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 206–8. http://dx.doi.org/10.1112/s1461157013000338.

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AbstractWe give a computationally effective criterion for determining whether a finite-index subgroup of $\mathrm{SL}_2(\mathbf{Z})$ is a congruence subgroup, extending earlier work of Hsu for subgroups of $\mathrm{PSL}_2(\mathbf{Z})$.

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47

Tkachenko, O. M. "Conditions, under which projective Sylow subgroups of almost locally normal group are locally conjugate." Researches in Mathematics 17 (January 29, 2021): 129. http://dx.doi.org/10.15421/240919.

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Two subclasses of expansions of locally normal groups with the help of finite p-groups are selected, at which projective Sylow p-subgroups are locally conjugate. The subgroup F of a finite index contains in the first case invariant Sylow p-subgroup, in the second case Sylow p-subgroups in F are almost abelian. The example that is given shows that for both subclasses the condition of expansion of locally normal group with the help of just p-group is essential.

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KIRSCHMER, MARKUS, and CHARLES LEEDHAM-GREEN. "COMPUTING WITH SUBGROUPS OF THE MODULAR GROUP." Glasgow Mathematical Journal 57, no. 1 (August 26, 2014): 173–80. http://dx.doi.org/10.1017/s0017089514000202.

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AbstractWe give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

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Bux, Kai-Uwe, and Cora Welsch. "Coset posets of infinite groups." Journal of Group Theory 23, no. 4 (July 1, 2020): 593–605. http://dx.doi.org/10.1515/jgth-2019-0162.

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AbstractWe consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

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CUTOLO, GIOVANNI, and HOWARD SMITH. "A NOTE ON POLYCYLIC RESIDUALLY FINITE-p GROUPS." Glasgow Mathematical Journal 52, no. 1 (December 4, 2009): 137–43. http://dx.doi.org/10.1017/s001708950999022x.

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AbstractA subgroup H of a residually finite-p group G is almost p-closed in G if H has finite p′-index in H, its closure with respect to the pro-p topology on G. We characterise polycyclic residually finite-p groups in which all subgroups are almost p-closed and discuss a few conditions that are sufficient for particular subgroups H to be almost p-closed. We also present, for each prime p, an example of a polycyclic residually-p group G for which |H: H| takes on all possible values, including infinity, as H varies.

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