A Characterization of Cyclic Groups via Indices of Maximal Subgroups

Maximal subgroups in torsion branch groups - arXiv

This allows for the easy construction of many previously unknown groups without maximal subgroups of infinite index. On the other hand, we ...

On groups with countably many maximal subgroups - De Gruyter

The next result is almost the only property shared by all CG-groups. Lemma 2. If G is a CG-group, every maximal subgroup of G has countable index. Proof. Let M ...

p-Groups with an Abelian Maximal Subgroup and Cyclic Center

They coincide with the class of nonabelian finite p-subgroups of GL(p, F), where F is a field, not of characteristic p, which contains all p power roots of 1, ...

Graph of Co-Maximal Subgroups in The Integer Modulo N Group

We explore the connectivity, patterns, and specific cases, offering a comprehensive analysis of this algebraic structure. Through a combination ...

Maximal subgroups of small index of finite almost simple groups

We prove in this paper that every almost simple group R with socle isomorphic to a simple group S possesses a conjugacy class of core-free ...

ON THE INTERSECTION NUMBER OF FINITE GROUPS

For each i ∈ {1,2,...,k}, the maximal subgroups of G with index pi are in ... maximal, cyclic subgroups of order p − 1. Since ... ) = ∞. The alternating group A6 ...

GAP (ref) - Chapter 39: Groups - Documentation

A conjugacy class is an external orbit (see ExternalOrbit (41.12-9)) of group elements with the group acting by conjugation on it. Thus element tests or ...

THE NORMAL INDEX OF A MAXIMAL SUBGROUP OF A FINITE ...

The result now follows by using inductive arguments and Lemma 2.3. Theorem 2.7 [9, Theorem 11]. Let G be a group and p ,q be two distinct primes dividing the ...

The Theory of p-Groups

Proof: Firstly notice that every maximal subgroup of a p-group is normal, and of index p. This means that if M is a maximal subgroup of G, then G/M is cyclic of ...

Extremely closed subgroupsand a variant on Glauberman's Z

Let G be a finite nonabelian simple group. Let M be a maximal subgroup of G and let H be a normal subgroup of M of prime index. Then. G = ⟨H ...

Verifying a Proof about Maximal Subgroups of Cyclic Groups

In summary, if a subgroup H is maximal in a cyclic group G of order n, then H is equal to for some prime p dividing n, ...

p-Groups - www-users.cs.umn.edu

Theorem 5.40. If G is a nilpotent group, then every maximal subgroup H is normal and has prime index. Proof. By Theorem 5.38 ...

Exponent-Critical Groups - arXiv

So at least one maximal subgroup of a non-abelian exponent-critical group must be abelian. But a finite group with an abelian maximal subgroup ...

CHARACTERIZATION OF FINITE DEDEKIND GROUPS

that 1-uniserial subgroups of G are normal as maximal subgroups of a ^-group. Every 2-uniserial subgroup of G has index p2 in G and such a subgroup certainly.

Computing the number of subgroups and normal, cyclic, maximal ...

The Dihedral group is defined as ha, b | an = b2 = e | bab = a−1i with subgroups hadi, ∀d|n and had,ajbi, ∀d|n, 1 ≤ j ≤ d. Likewise, the cyclic group. C2 ...

The Nonabelian Simple Groups $G, |G| < 10^6$--Maximal Subgroups

(m, n)+ is the even subgroup of index two in Em X En ... We do an analysis of the maximal subgroups of the ... H. H. MITCHELL, "The subgroups of the quaternary ...

The subgroup structure of finite classical groups in terms of ... - AMiner

The stabilizer of an imaginary triangle is the normalizer of a Singer cyclic subgroup. Theorem 2.5 (Hartley) When q is even, the maximal subgroups of PSL3(q) ...

SUBGROUP SERIES II 1. Introduction In part I, we met nilpotent and ...

This p-power is p itself because maximal subgroups in a supersolvable group have prime index. This proof of Corollary 4.19 lies a little deeper than the ...

List of problems on discrete subgroups of Lie groups and their ...

A subgroup H of a group G is called maximal if there is no proper subgroup between H and G. Some maximal subgroups have finite index in G (for instance, ...

(IUCr) A simple characterization of the subgroups of space groups

The subgroups of finite index of any n-dimensional space group are determined by the solutions of a set of congruences analogous in form and meaning to the ...