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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2
votes
1answer
371 views

Why is this subgroup normal?

Let $G$ be a group of odd order and $H$ a subgroup of index 5, then $H$ is normal. How can I prove this? (there is an hint: use the fact that $S_5$ has no element of order $15$) I took the usual ...
6
votes
1answer
2k views

Converse of Lagrange's theorem for abelian groups

I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup ...
7
votes
1answer
289 views

Reference for classification of small groups

There are various online resources for the classification of groups of small order, such as this one or that one. Is there any nice reference in the literature which contains such a classification ...
5
votes
4answers
332 views

About the Order of Groups

Is there any theorem that states the all the finite groups of order n are the same? or some sort of theorem that refers to the order of two finite groups? If anyone can post a reference to this topic ...
3
votes
1answer
75 views

Why are generators of $Z^{*}_p, p=c \cdot 2^k + 1$ so small?

I was implementing NTT for long integer multiplication and thus searched for generators of several $Z^{*}_p$ groups where $p=c\cdot 2^k + 1$. I used the algorithm described in Wikipedia which uses ...
1
vote
1answer
84 views

Infinitely many nilpotent elements in $\mathbb{C}[G]$

Suppose $G$ is a finite group and $F$ is a field such that $\mathrm{char}\;F$ doesn't divide $|G|$. Suppose that $F$ is algebraically closed and $G$ is not abelian. How can I prove that $F[G]$ has ...
4
votes
1answer
216 views

Primitive root modulo $p$

I am currently trying to find a primitive element of the multiplicative group of field $GF(p)$. Since the numbers are relatively small, I know the factorization of $$\phi(p)=p-1 = {p_1}^{k_1} ...
3
votes
1answer
147 views

Why is it that any two distinct subgroups of G of order p (prime) intersect in 1?

Why is it that any two distinct subgroups of G of order p (prime) intersect in 1? It says so here on page 29. But why is it?
1
vote
0answers
95 views

Is it true that if $g^{-1} = g$, then $g$ is only conjugate to itself in a finite group $G$?

I have a finite group G, where $Z(G)=1$, and I have an element $g\neq 1$ where $g^{-1} = g$ and $gg=1$. I want to say that $g$ is then only conjugate to itself so that I have a contradiction ($Z(G)$ ...
3
votes
1answer
160 views

Is my textbook wrong about this corollary of Sylow's theorem?

Is my textbook wrong about this corollary of Sylow's theorem? Let $G$ be a finite group and $p$ a prime that divides $|G|$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Then $n_p = 1$ if ...
2
votes
1answer
117 views

Group acting transitively on subset and what must hold for that subset

I've been struggling with this problem for some time now so any help would be greatly appreciated. I have a finite group $G$ and a subset $A$ of $G$. I am told that $G$ acts transitively on $A$ and ...
1
vote
1answer
842 views

Prove that every p-group is nilpotent.

Let $G$ be a p-group $|G| = p^n$. Then G is nilpotent in the following sense: Let $C^1(G) = [G,G] = G'$ And $C^n(G) = [G,C^{n-1}]$. G is nilpotent iff there is a $n_0$ such that $C^n = \{e\}$. What ...
0
votes
1answer
45 views

Finite group - binary

Prove that $g(\alpha)$=0 if and only if $g'(\alpha)$=0 $g(t)=t^{11}+t^{10}+t^6+t^5+t^4+t^2+1$ $g'(t)=t^{11}+t^9+t^7+t^6+t^5+t+1$ where $\alpha \in F[t]$. We are working in standard binary space. ...
12
votes
2answers
556 views

Is there an infinite simple group with no element of order $2$?

According to the Feit-Thompson theorem, every group of odd order is solvable and thus every finite nonabelian simple group has even order. Thus every finite nonabelian simple group has an involution ...
4
votes
1answer
110 views

The order of the orbits of the action of a p-subgroup on set of left cosets of a Sylow p-subgroup

If $H$ is a $p$-subgroup of $G$ with order of $p^{a}$ and $K$ is a Sylow $p$-subgroup of $G$ with order of $p^{b}$. $X$ is the set of left coset of $K$. Let $H$ acts on $X$, what is the order of ...
1
vote
1answer
126 views

The normality and intersection of two sylow group

$G$ is of order 1225. According to the Sylow theorem, we know that $H$ is the only sylow 5-subgroup of $G$ and $K$ is the only sylow 7-subgroup of $G$. I am trying to prove that they are both normal ...
4
votes
2answers
2k views

A group of order $p^2q^2$ is never simple

Let $p,q$ be primes and let $G$ be a group of order $p^2q^2$, what's the best way to show $G$ is non-simple? I know it suffices to show that one of the sylow-p or sylow-q subgroup of $G$ is normal, ...
1
vote
1answer
714 views

A question about the fixed point and group action.

A group $G$ of order $35$ act on a set $S$ that has $16$ elements. Must the action have a fixed point? Why?
2
votes
2answers
761 views

How does one decompose the regular representation of $S_3$?

I need to decompose the regular representation of $S_3$ into irreducible ones. What I know so far is this: $S_3$ is generated by $\tau = (12)$ and $\sigma = (123)$. If $v$ is an eigenvector of ...
2
votes
1answer
79 views

Induced automorphisms of regular automorphisms on factor groups

If you have a $\phi$-invariant, normal subgroup $N$ (so $\phi(N)=N$) of a finite group $G$, for an $\phi$, then you get an induced automorphism of $G/N$ by $gN\mapsto \phi(gN)=\phi(g)N$. The order of ...
2
votes
1answer
74 views

Ring structure of non-modular group cohomology

I know if $k$ is a field such that $char(k)$ divides $|G|$ (a finite group), then finding the ring structure on $H^\ast(G,k)$ can be very, very hard. But what about when $k=\mathbb{Z}$? Is the ...
3
votes
1answer
101 views

Embedding of a finite abelian group in $(\mathbb{Z}_n)^*$

Let $G$ be a finite abelian group. How do I show that $G$ can be embedded in $(\mathbb{Z}_n)^*$ for some n? Also, that there exists a subgroup $H$ such that $G$ is isomorphic to ...
3
votes
1answer
174 views

Simplify the category of finite abelian groups

Consider the category $\mathsf{FinAb}$ of finite abelian groups. The structure theorem tells us that we can write down a skeleton for this category (a set of representatives for the isomorphism ...
0
votes
1answer
83 views

How does one show that $(n-2)! = 2^{m-1} m! (n - 2m)!$ has only one solution for $n\ne 6$?

The obvious solution is 1, for $n=6$ there is another one - $m=3$. How does one show that for other $n$ there are no solutions but $m=1$? This is to show that for $n\ne 6$ all automorphisms of $S_n$ ...
2
votes
1answer
86 views

Explanation of $|G|=d_1 n_1^2 + \cdots +d_s n_s^2$ in representation theory

Can anyone help me to understand the equation $$|G|=\sum^s_{i=1} d_i n_i^2$$ please? The context is representation theory of finite groups over a field $\mathbb{K}$ of characteristic zero. I know ...
11
votes
2answers
252 views

How many automorphisms of $S_n$ take transpositions into transpositions?

I need to show that an automorphism of $S_n$ which takes transpositions to transpositions is an inner automorphism. I thought it could be done by showing that such automorphisms form a subgroups ...
7
votes
1answer
297 views

Left and right transversals of groups.

I have recently been looking at Hall's marriage theorem. One application of it is that given a finite group $G$ and a subgroup $H\leq G$, there is a left transversal of $H$ that is also a right ...
2
votes
1answer
102 views

Number of elements of order 9 and 6 in $C_4 \times C_9 \times C_3 \times C_3$

I need to find the number of elements of order 9 and 6 in the group $C_4 \times C_9 \times C_3 \times C_3$. I have read the solution and they are supposed to be 54 elements of order 9 and 26 elements ...
5
votes
1answer
293 views

First publication of Pierre Deligne

The first publication of Pierre Deligne was Congruences sur le nombre de sous-groupes d’ordre $p^k$ dans un groupe fini, Bull. Soc. Math. Belg. XVIII 2 (1966) pp. 129–132. I do not have access to this ...
0
votes
1answer
280 views

does the trivial representation always induce the permutation representation?

Does the trivial representation always induce the permutation representation? Is this true for each field $\mathbb{K}$ or just for representations over $\mathbb{C}$?
1
vote
4answers
129 views

Problem of finding subgroup without Sylow's Thm.

Let $G$ is a group with order $p^n$ where $p$ is prime and $n \geq 3$. By Sylow's Thm, we know that $G$ has a subgroup with order $p^2$. But, I wonder to proof without Sylow's Thm.
0
votes
1answer
227 views

induction method

In Theorem 3.1 in this paper, part (1) of the proof, $X$ may not be a subgroup of $N$, so how did the authors apply the induction method? Also, I do not understand this: In the proof of Theorem 3.2 ...
2
votes
2answers
270 views

Is $A_n$ characteristic in $S_n$?

The title is the question. Is $A_n$ characteristic in $S_n$? If $\phi \in \operatorname{Aut}(S_n)$, Then $[S_n : \phi(A_n)]$ (The index of $\phi(A_n)$) is 2. Maybe the only subgroup of $S_n$ of ...
6
votes
1answer
4k views

Computing the Smith Normal Form

This question is related to the Smith Normal Form of Matrices: Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $R :=$ $$ \begin{bmatrix} -6 & 111 & ...
2
votes
1answer
172 views

Sylow subgroups of a group G and product of subgroups

Let $G$ be a finite group. Let $H \leq K\unlhd G$. If for each $P$ Sylow subgroup of G there exists $x \in G$ such that $HP^{x}=P^{x}H$ then for each $P \cap K$ of $K$ there exists $y \in K$ such that ...
8
votes
0answers
212 views

Generating the partners in a multi-dimensional irreducible representation.

I am trying to block diagonalize a Hermitian matrix using the irreducible representations of its symmetry group. Using the group's character table, it is straightforward to generate a set of ...
1
vote
1answer
152 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
4
votes
1answer
1k views

Dedekind modular law

Dedekind modular law. If $A,B,C$ are subgroups of a group $G$ with $A \subseteq B$ then $A(B \cap C) = B \cap AC$. Below is what I want to prove. Let K be a finite group with $K = LH$, where $L,H$ ...
0
votes
1answer
176 views

Sylow theorem and icosahedral group

I think this might be a stupid question. The icosahedral group $A_5$ with order 60 is a simple group $60=2^2\times 3\times5$ but according to Sylow theorem $A_5$ must have subgroups of order 4. But ...
1
vote
2answers
324 views

P-groups,chain of subgroups of index p

Let $G$ be a $p$-group. Let $H$ be any subgroup of $G$. How to prove that there exists subgroups of $G$ such that $$H = H_0 \lt H_1 \lt H_2\lt \cdots \lt H_n=G$$ such that $|H_{i+1}/H_i|=p$? I have ...
2
votes
1answer
201 views

Why are Sylow 2-subgroups of $S_n$ self-normalizing?

Why are Sylow 2-subgroups of $S_n$ self-normalizing? I've read before that this is the case but haven't seen a proof.
1
vote
2answers
157 views

Injectivity of an endomorphism

I am having trouble with showing that the function in this problem is injective. I've been trying it for a while already. Surjectivity wasn't hard, and neither was proving that it was a homomorphism. ...
3
votes
1answer
169 views

Sylow subgroups of a group G

If $G$ is a finite group and $P$ is a Sylow $p$-subgroup of with $P=HK$, $H$,$K$ are subgroups of $P$ and if $Q$ is a Sylow $q$-subgroup of $G$ and $H^{a}Q=QH^{a}$, $K^{b}Q=QK^{b}$ for some $a,b \in ...
3
votes
0answers
232 views

On Sylow subgroup of simple group PSL(2,p)

Let $G$ be finite group such that $Z(G)=1$ and the number of Sylow $p$-subgroups of $G$ equal to the number of Sylow $p-$subgroups of $PSL(2,r)$ for every prime $p$ ( $r$ is prime and $r^2$ not ...
1
vote
1answer
173 views

Why all the elements of sylow subgroups are not adding up to the no. of elements of Group.

Show that a group of order 70 can not be simple. I've tried to solve using Sylow theorem. I got 1, 5, 7, 35 Sylow 2-subgroups, 1 sylow 5-subgroup and 1 sylow 7-subgroup. Now the only choice is 35 ...
0
votes
2answers
92 views

Finding $\mathbb{Z}_{p^{3}}(p)$

I have to find $\mathbb{Z}_{p^{3}}(p)$ (p is a prime). I know that the order of any element in $\mathbb{Z}_{p^{3}}$ divides $p^{3}$. How can I then deduce that ...
5
votes
3answers
546 views

Platonic Solids

It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses ...
1
vote
0answers
113 views

Hall subgroup of a finite group G

If $G$ is a finite group and $x$ is a $p'$ element of G does this imply that there a Hall $p'$ subgroup of $G$ containing $x$?
7
votes
0answers
296 views

What groups are semidirect products of simple groups?

The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups groups, and is closed under ...
3
votes
0answers
166 views

About a Sylow subgroup of a product

Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of G such that $P=(P\cap H)(P\cap K)$. I looked at the proof here, but I ...