Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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4
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3answers
622 views

A finite abelian group whose order is divisible by 10 contains an element of order 10

It is given that the order of some finite abelian group is divisible by 10. Prove that the group has a cyclic subgroup of order 10. It is clear that since order of group is divisible by 10. By ...
11
votes
2answers
276 views

Embeddings of finite groups into $\mathrm{GL}_n(\mathbb{Z})$

We know that every group $G$ of order $n$ can be embedded into $\mathrm{GL}_n(\mathbb{Z})$, because $G \hookrightarrow S_n$ (Cayley theorem) $S_n\hookrightarrow \mathrm{GL}_n(\mathbb{Z})$ (via ...
3
votes
2answers
977 views

Some questions on the Frattini subgroup

According to Derek J.S Robinson's A Course in the Theory of Groups, the Frattini subgroup of a group $G$, denoted $\mathrm{Frat}G$ is defined to be the intersection of all maximal subgroups of $G$. ...
2
votes
3answers
330 views

A simple question on characteristic subgroups

Suppose $P$ is a finite abelian $2$-group and $U$,$V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?
3
votes
0answers
138 views

A class of finite groups

Let $G$ be a finite group having the property that for any prime $p$ dividing $|G|$, it has a subgroup H with $[G:H]=p$. What can be said about these groups? I believe I can prove that they must be ...
7
votes
1answer
130 views

2-transitive action on vector space of char. 2

Suppose $V$ is an elementary abelian 2-group of order $2^k$, with $k>2$. Let $H\le GL(k,2)$ be a solvable group of automorphisms of $V$. How does one prove that $H$ cannot act 2-transitively on ...
3
votes
1answer
94 views

Does this group action fix a point?

I saw a question on a forum earlier, and I'm curious to see how it can be solved. Suppose that $G$ is a group of order $pr$ for distinct primes $p$ and $r$, and let $G$ act on a set $S$ of order ...
4
votes
4answers
154 views

Showing a normal subgroup contains a subgroup

Let G be a finite group, $H \le G$, and $N\lhd G$. Suppose $|H|$ and $|G :N|$ are relatively prime. Is it true that $H \le N$? Since $N$ is a normal subgroup, I know that $NH \le G \implies |NH|$ ...
2
votes
1answer
70 views

Order of an element in the group $(\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times}$

For p a prime and n a positive integer, consider the group of units, $(\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times}$. How can I go about to find the order of $\bar{p}$?
1
vote
5answers
325 views

A criterion for a group to be abelian

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let and ...
7
votes
0answers
185 views

A problem in transfer theory

This is about problem 5B.1 page 157 in Isaacs' "Finite Group Theory" book. This chapter is definitely giving me trouble. The problem reads: Let $G$ be a finite group, $P \in \operatorname{Syl}_p(G)$ ...
2
votes
0answers
163 views

If $G$ is a finite group and $(ab)^3= a^3b^3$, [duplicate]

Possible Duplicate: Group with an endomorphism that is “almost” abelian is abelian If $G$ is a finite group and $(ab)^3= a^3b^3$, and $3 \nmid o(G)$, then how do I prove that ...
15
votes
4answers
3k views

Group of order 15 is abelian

How do I prove group of order 15 is abelian? Is there any general strategy to prove that a group of particular order(composite order) is abelian?
2
votes
2answers
2k views

Order of product of two elements in a group

Let $G$ be an abelian group. Let $a, b \in G $ and let their order be $m$ and $n$ be respectively. Is it always true that order of $ab$ is $lcm(m,n)$? What if $m$ and $n$ are coprime to each other?
8
votes
3answers
576 views

Square free finite abelian group is cyclic

How do I show every abelian group whose order is square free is cyclic without using the fundamental theorem of finite abelian groups? I tried something like this Let $|G| = p_1p_2...p_n$ By ...
10
votes
2answers
245 views

Finite/Infinite Coxeter Groups

In the same contest as this we got the following problem: We are given a language with only three letters letters $A,B,C$. Two words are equivalent if they can be transformed from one another ...
8
votes
2answers
973 views

Which $p$-groups can be Sylow p-subgroups with trivial intersection?

Every cyclic p-group is a Sylow p-subgroup of a finite group whose distinct Sylow p-subgroups intersect trivially in pairs (and there is at least one pair). For instance, let q be a prime congruent ...
3
votes
1answer
328 views

Finite Subgroups of general linear group

If $H$ is a finite subgroup of $GL(n,\mathbb{Z})$ then by Minkowskie's theorem, it injects to a subgroup of $GL(n,\mathbb{Z}/p\mathbb{Z})$ under the natural map from $GL(n,\mathbb{Z})$ to ...
4
votes
2answers
123 views

Conjugacy of projective representations

Given characters of the Schur covering group of $G$ of the same degree, how does one tell if the projective representations (as homomorphisms from $G$ into $\operatorname{PGL}$) are conjugate in ...
19
votes
1answer
314 views

Finite Groups with a subgroup of every possible index

Suppose $G$ is a finite group, with $|G|=n$. Suppose also that for every positive integer $m\mid n$, $G$ has a subgroup of index $m$. Are there any general statements (structural or otherwise) I can ...
9
votes
2answers
261 views

If $G/Z$ is a product of $p$-groups…

Let $G$ be a finite group. Suppose $Z \le Z(G)$ and $G/Z \cong A\times B$, where $A$ is a $\pi$-group and $B$ is a $\pi'$-group for some set of primes $\pi$. Is it true that $G\cong C \times D$, ...
17
votes
4answers
3k views

How does one compute the sign of a permutation?

The sign of a permutation $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula $${\rm sgn}(\sigma) ...
2
votes
1answer
80 views

Non conjugate $p$-subgroups of $\mathrm{GL}_n(\mathbb Z)$

It is probably not that difficult but I can't find an example of two non-conjugate $p$-subgroups (of same order) of $\mathrm{GL}_n(\mathbb Z)$ ($n>1$).
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vote
2answers
547 views

Groups of order 12 without Sylow

It is clear that Sylow theorems are an essential tool for the classification of finite groups. I recently read an article by Marcel Wild, The Groups of Order Sixteen Made Easy, where he gives a ...
11
votes
1answer
723 views

$A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ only simple group of order 360

How can one show, without the use of character theory, that $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ is, up to isomorphism, the only simple group of order 360?
4
votes
2answers
252 views

Check whether two subgroup of $GL(n,\mathbb Z)$ are conjugate

Suppose I have two finite subgroups of $GL(n,\mathbb Z)$. Is there an algorithm to find out whether these two belong to the same conjugacy class in $GL(n,\mathbb Z)$? I tried by using the Jordan ...
17
votes
2answers
937 views

exercise in Isaacs's book on Character Theory

I'm stuck on an exercise in Isaacs's book "Character Theory of Finite Groups" - it relates to something I'm looking at as part of ongoing research, but I guess it belongs here rather than on ...
2
votes
1answer
92 views

Subgroup relations in $GL(3,\mathbb Z)$

There are 73 conjugacy classes of finite subgroups in $GL(3,\mathbb Z)$. If you take 73 representatives, you will find group-subgroup relations between them. There must exist an overview from the ...
6
votes
1answer
192 views

Maximal Subgroups and order of a group

I encountered the following exercise in Isaacs' Algebra: "Suppose a group $G$ has only one maximal subgroup. Prove that the order of $G$ must be a power of a prime". I think I've proven this for the ...
4
votes
1answer
128 views

Is there a general result that groups of order $2^n\cdot 3$ are solvable?

For the past day or so I've been trying to solve an exercise in Lang showing all groups of order less than $60$ are solvable. Excluding the case of order $56$, most cases are taken care of by other ...
2
votes
0answers
106 views

a question about elements of permutation groups

Let $n \in \mathbb{N}$ and let $S_n$ denote the permutation group on $n$ letters. I'm trying to figure out what kind of elements $\sigma, \delta \in S_n$ satisfy the following relations: ...
3
votes
0answers
107 views

Blocks and covering modules

I have a question relating to p109 of Local representation theory by JL Alperin. Let $G$ be a finite group and let $N$ be a normal subgroup. If $B$ is a block of $G$, why must $B$ be a summand of ...
1
vote
2answers
419 views

Problem from Armstrong's book, “Groups and Symmetry”

I haven't gotten all that far with this: If $a$, $b$ are members of the permutation group $S_n$, and $ab=ba$, prove that $b$ permutes those integers which are left fixed by $a$. Show that $b$ must ...
1
vote
1answer
52 views

Blocks and simple modules

I have a (probably very straightforward) question about blocks and simple modules. The problem I'm having is on p103 of Local representation theory by JL Alperin. Let $G$ be a finite group. Let $B$ ...
3
votes
1answer
182 views

Mackey and relatively projective modules

While reading over Alperin's Local Representation Theory and reminding myself how a module is relatively H-projective iff H contains some vertex of the module, I realized I could not prove a basic ...
3
votes
1answer
74 views

Is relatively free the same thing as induced for finite group modules?

I was looking over Alperin's Local Representation Theory and I realized I remembered a definition that may not be there (or true). Is a relatively H-free G-module exactly the same as a G-module ...
2
votes
1answer
81 views

Defect groups and subgroups

I would like to prove the following statement from Alperin's Local representation theory, p101: Lemma Let $b$ be a block of the subgroup $H$ of $G$ and let $D$ be a defect group of $b$. If $b^G$ is ...
1
vote
1answer
293 views

Sylow theorems and normalizer

I'll state the result I'm trying to prove, progress I've made, and the two questions I have which will help me solve it. The question is originally motivated by studying defect groups in modular ...
6
votes
1answer
370 views

Semidirect product uniqueness argument for classifying groups of small order

I'm having trouble understanding the following method for determining the number of semidirect products between two groups in simple cases that arise when trying to classify certain groups of small ...
7
votes
3answers
914 views

Three finite groups with the same numbers of elements of each order

There exist pairs of finite groups $G$ and $H$ such that $G$ and $H$ are not isomorphic, yet they have the same number of elements of each order. For example, if $p$ is an odd prime, then the group ...
4
votes
3answers
227 views

Is there an elegant way to determine which subgroups of $S_3$ are normal?

I have a homework problem which reads List all subgroups of $S_3$ and determine which subgroups are normal and which are not normal. I understand the definitions of subgroup and normal subgroup, ...
1
vote
1answer
92 views

Clarifications on proof that the fixed points of order $p$, $i_p(G)\equiv -1\pmod{p}$

I'm reading this paper by Marcel Herzog on jstor: http://www.jstor.org/stable/2040939?seq=1 I want to follow up on a few things about the short proof of Theorem 1, found on the bottom of page 1 of ...
9
votes
3answers
351 views

Showing $H\unlhd G$ when $[G\colon H]$ is not the smallest prime dividing $|G|$

I recently read about the theorem that for a finite group $G$, if $p$ is the least prime dividing $|G|$, then any subgroup $H$ with $[G\colon H]=p$ is normal in $G$. Going over some exercises, this ...
2
votes
0answers
113 views

What is $\operatorname{Aut}(\operatorname{PSL}_2(\mathbb{F}_q))$? [duplicate]

Possible Duplicate: Automorphisms of projective special linear group I'm sure this is well known, but I don't know where to look up such things. What is ...
1
vote
1answer
99 views

Alperin p82 Lemma 11.3

My question is regarding Lemma 11.3 on p82 of Local representation theory by JL Alperin; the Google Books preview unfortunately does not contain this page. I need to prove the following claim: Claim: ...
3
votes
4answers
396 views

Quaternion group as an extension

I'm trying to understand how the quaternion group Q arises as an extension of $\mathbb{Z}_{4}$ by $\mathbb{Z}_{2}$. More precisely, I'm trying to find the two homomorphisms in the short exact sequence ...
2
votes
1answer
189 views

Alperin “Local Representation Theory” Lemma 9.7

Have a question about the proof of Lemma 9.7 from Alperin "Local Representation Theory", p69: Lemma 7 If $U$ is an indecomposable $kG$-module with vertex $Q$ and trivial source and $H$ is any ...
3
votes
3answers
354 views

Geometrical meaning of automorphisms of cyclic groups

I'm looking for a geometrical interpretation of the action of automorphisms of cyclic groups. I'll take one particular example to make it clear : I'm taking the cyclic group $\mathbb{Z}_{12}$, which ...
4
votes
3answers
1k views

Finite Subgroups of GL(n,R)

A nice result about $GL(n,\mathbb{Z})$ is that it has finitely many finite subgroups upto isomorphism; and also any finite subgroup of $GL(n,\mathbb{Q})$ is conjugate to a subgroup of ...
3
votes
2answers
177 views

Non-commuting coprime elements in finite non-abelian groups

Let G be a finite non-abelian group having two distinct primes dividing its order. Is it always true that G has two non-commuting Sylow subgroups? $\hskip300pt$ Thank you. EDIT: I asked in a comment ...