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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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 $\mathrm{GL}_n(\mathbb Z)$ ($n>1$).
1
vote
2answers
545 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
931 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
418 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
179 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
73 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
905 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
394 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
350 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
995 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 ...
2
votes
3answers
252 views

Subgroup of a Finite Group

Let $G$ be a finite group, and let $S$ be a nonempty subset of $G$. Suppose $S$ is closed with respect to multiplication. Prove that $S$ is a subgroup of $G$. (Hint: It remains to prove that $S$ ...
2
votes
3answers
295 views

Normal subgroups vs characteristic subgroups

It is well known that characteristic subgroups of a group $G$ are normal. Is the converse true?
1
vote
1answer
596 views

Using order to show isomorphism in a finite abelian group

How can I show that two finite abelian groups are isomorphic, knowing only that both groups have the same number of elements of any given order? I feel like there should be a nice way to show this ...
13
votes
2answers
900 views

Application of the Sylow Theorems to groups of order $p^2q$

I am trying to show that any group of order $p^2q$ has a normal Sylow subgroup where $p$ and $q$ are distinct primes. In the case $p>q$ I have no problem.. By Sylow $n_p|q$, so $n_p$ is either $1$ ...
5
votes
1answer
2k views

No group of order 36 is simple

Fraleigh(7ed) Example37.14 No group of order 36 is simple. Such a group $G$ has either $1$ or $4$ subgroups of order $9$. If there is only one such subgroup, it is normal in $G$. If there are four ...
8
votes
1answer
319 views

$PSL(3,4)$ has no element of order $15$

$PSL(3,4)$ has no element of order $15$. Thus it is no isomorphic to $A_8$. Here, $PSL(3,4)$ denotes the $3 \times 3$ projective special linear group on the field with $4$ elements. As listing ...
6
votes
2answers
238 views

Why does the number of orbits of the stabilizer of an element in a transitive group not depend on the element?

Let $E$ be a finite set and $G$ a transitive group of permutations of $E$. For $x\in E$, let $S_x$ denote the stabilizer of $x$. Then, for any $x,y\in E$, the number of orbits of $E$ under the action ...
3
votes
1answer
106 views

A subgroup which is generated by $W$-marginal subgroups is itself $W$-marginal

There is a conclusion on $W$-marginal subgroups of a group: A subgroup which is generated by $W$-marginal subgroups is itself $W$-marginal. Here, $W$ is a set of words. In a group $G$, a normal ...
5
votes
3answers
305 views

Roots of $x^2 + 2x + 2$

I'm trying to show that there are infinitely many values of $p$ such that $x^2 + 2x + 2$ has no roots over $\mathbb{F}_p$. Is this easily solvable? (I kind of came up with it myself so I don't ...
4
votes
1answer
249 views

Group of order $3^a\cdot5\cdot11$ has a normal Sylow 3-group

A group of order $3^a\cdot 5\cdot 11$ has a normal Sylow $3$-subgroup. This is question 5C.7 in Isaacs's Finite Group Theory. That section of the text is about transfer in finite groups, and ...
10
votes
2answers
727 views

If $|H|$ and $[G:K]$ are relatively prime, then $H \leq K$

Let $H$ and $K$ be subgroups of a finite group $G$, at least one of which is normal. Show that if $|H|$ and $[G:K]$ are relatively prime, then $H \leq K$. In the case that $K$ is normal, let $\pi : G ...
4
votes
2answers
295 views

Isomorphism between group algebras

I am starting to study group algebras and I am stuck on the following problem. The first part is easy, but I copy it in case it helps to prove the second part. This exercise is taken from ...
2
votes
1answer
330 views

Additive Cyclic Group

I'm trying to find an element $k$ that generates the cyclic additive group $\mathbb{Z}_{6}$. Since a group is cyclic, the entire group can be generated by a single element. I've tried adding ...
3
votes
3answers
251 views

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...
5
votes
4answers
327 views

A common mistake (?) on group algebras

I have just started studying group representations with the book Representations of Groups by Lux and Pahlings (published by Cambridge). I have tried to solve some exercises to understand the concept ...
2
votes
3answers
162 views

Problems involving a finite abelian group

I'm trying to prove the following statements. Let $G$ be a finite abelian group $G = \{a_{1}, a_{2}, ..., a_{n}\}$. If there is no element $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} ...
11
votes
5answers
573 views

Finite Abelian Group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2}$ = $e$. I've been stuck on this problem for quite some time. Could ...
4
votes
1answer
266 views

Representation of Finite Groups

Is it true that any finite group determined by representation over closed field? In other words, are there exists two different groups with the same representations? For example, any non-abelian group ...
23
votes
4answers
2k views

Finite Groups with exactly $n$ conjugacy classes $(n=2,3,…)$

I am looking to classify (up to isomorphism) those finite groups $G$ with exactly 2 conjugacy classes. If $G$ is abelian, then each element forms its own conjugacy class, so only the cyclic group of ...
2
votes
3answers
128 views

Finding a (small) prime great enough that there are at least m elements of order m

I'm hoping that someone can provide me with some results or point me in the right direction. I'm working with finite fields; really, I'm just doing arithmetic modulo a prime $p$. I'm taking elements ...
3
votes
0answers
213 views

[H,K] abelian if K centralizes [H,K]

Here is a paraphrased version of problem 4B.4 in Isaacs's Finite Group Theory: Let $G$ be a group and $X,Y$ subgroups of $G$, such that $Y$ centralizes $[X,Y]$. If $X$ is normal in $G$, show ...