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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4answers
84 views

$(p,\frac{n}{p^\alpha})=1$ then $p\nmid\binom{n}{p^\alpha}$

Let $n=p^\alpha m$ wherein $(p,m)=1$. Then we have $$p\nmid\binom{n}{p^\alpha}$$ What I have done is just playing with $\binom{n}{p^\alpha}$ ...
3
votes
2answers
288 views

An explicit calculation of Galois group

This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...
7
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1answer
462 views

Calculating the group co-homology of the symmetric group $S_3$ with integer coefficients.

I have been trying for a while to make sense of Ex V.3.5 & Ex III.10.1 in Brown's book 'Co-homology of Groups': Calculate the Co-homology of $S_3$ with co-efficients in $\mathbb{Z}$, possibly ...
6
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0answers
298 views

Resource: Group Theory

There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory. I want to see the Ph.D. thesis of Raymond T. Shepherd, ...
3
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2answers
64 views

$P\in \operatorname{Syl}_p(S_n)$ implies that $P\in \operatorname{Syl}_p(A_n)$ and $|N_{A_n}(P)|=\frac{1}{2}|N_{S_n}(P)|$

$\newcommand{\Syl}{\operatorname{Syl}}$ This is an exercise (with hint about the second part) in my own language book in Group theory, however, maybe it is a lemma or theorem in an standard book ...
2
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1answer
59 views

The $p^k$-rank of a subgroup is no greater than the $p^k$-rank of the group.

Recently I was given a handout containing (roughly) the following text: Let $A$ be a finite abelian group, and $p^k$ a prime power. The $p^k$-rank of $A$ is defined to be ...
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1answer
56 views

What is subgroup $\langle g^d \rangle$?

According to the introductory abstract algebra, it says that $g^n =e$ where $g$ is an element of some finite group. Then, it talks about the subgroup $\langle g^d \rangle$. What is it exactly, and ...
4
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1answer
247 views

normal p-subgroups of a finite group and chief factor

Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper ...
3
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1answer
275 views

exponent of an abelian group

Let $p$ be a prime. Let $H_{i}, i=1,...,n$ be normal subgroups of a finite group $G$. I want to prove the following: If $G/H_{i}$, $i=1,...,n$ are abelian groups of exponent dividing $p−1$, then $G/N$ ...
4
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1answer
239 views

Abelian subgroup of a group of order $2002$

Another unsolved question from my studying for quals - Show that if $G$ is a group of order $2002=2\cdot 7 \cdot 11 \cdot 13$, then $G$ has an abelian subgroup of index 2. I know it has to do with ...
5
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1answer
214 views

Complexity of finite group isomorphism problem

Consider the next decision problem: Given two finite groups represented by their multiplicity table, determine if they are isomorphic or not. Clearly, this problem belongs to NP since given a witness ...
4
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0answers
94 views

In finite groups does counting orders of elements is enough to determine if they are isomorphic [duplicate]

Possible Duplicate: Three finite groups with the same numbers of elements of each order Suppose that we have two finite groups $G$ and $H$ such that for each $n\in\mathbb{N}$ ...
8
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2answers
317 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
3
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1answer
248 views

Contragradient representation of a finite group

I am reading Serre's Linear Representations of Finite Groups and in an exercise in there he asks to show if $\rho$ is a representation of a finite group on $\textrm{GL}(V)$ with $V$ a finite ...
4
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1answer
569 views

Group cohomology of finite groups

I wonder if the group cohomology of a finite group $G$ with coefficients in $\mathbb{Z}$ is finite. This statement may be too strong. I am interested in, for instance, dihedral group. $$ ...
6
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1answer
181 views

Uniqueness of conjugates of a subgroup.

This question is partly influenced by the question: Are two subgroups that contain a common element conjugate iff they are conjugate under the normalizer? If we have an arbitrary finite group $G$ ...
0
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1answer
292 views

How to find all the subgroups of a group that contain a given subset?

Given a group G of order n and a subset S of G such that |S|=m. What is the best algorithm for generating all the subgroups of G that contain S? How the complextity of such algorithm depends on n and ...
6
votes
1answer
156 views

Group Theory Automorphism question

The question at hand is: Let G be a finite group and $\alpha$ an involutory automorphism of G, which doesn't fixate any element aside from the trivial one. 1) Prove that $ g \mapsto g^{-1}g^{\alpha} ...
5
votes
2answers
139 views

Group with an automorphism of order 2 (Jacobson BA1)

I am having trouble with Exercise 11, Section 1.10 of Basic Algebra 1 by Nathan Jacobson (pub. Freeman & Co. 1985). The statement to prove is: Let $G$ be a finite group and $\phi$ an ...
5
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1answer
997 views

A group of order $120$ cannot be simple

We know that: Theorem: If a simple group $G$ has a proper subgroup $H$ such that $[G:H]=n$ then $G\hookrightarrow A_n$. This fact can help us to prove that any group of $G$ of order $120$ is not ...
7
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1answer
418 views

Classifing groups of order 56: problems with the semidirect product

While I was doing an exercise about the classification of groups of order 56, I had some problems concerning the semidirect product. Let $G$ a group of order 56 and let us suppose that the 7-Sylow is ...
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1answer
109 views

The classification of irreducible representations of finite and compact Lie groups over $\mathbb{C}$

I just started reading a few lecture notes on representation theory and I was wondering about a big picture that one should keep in mind while reading through these lecture notes. Have all ...
2
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0answers
125 views

Conjugacy classes in $SL(n,q)$

If $q$ is a prime power, then the conjugacy classes in $GL(n,q)$ are determined completely by minimal polynomials. Now in the subgroup $SL(n,q)$, these conjugacy classes can split, in the sense they ...
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1answer
111 views

Why is the Wedderburn formula in this case wrong?

in this question counterexample: degree of representation $\leq$ index of normal subgroup there was the answer (in the second comment under the answer), that the dihedral group $D_5$ hat exactly 3 ...
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2answers
34 views

Question on trace-weighted sums for irrep of finite group

For a finite group $G$, is the following true, where $\rho$ is a finite-dimensional complex unitary irreducible representation? $$\sum _{g \in G} \mathrm{Tr} (\rho(g)) \rho(g) = \frac{|G|}{n} ...
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0answers
117 views

Unique minimal normal subgroup

I'm referring to this post: Unique minimal normal subgroup $\implies$ faithful irreducible representation. Isn't the claim, that there is always a faithful irreducible representation (if $char(K) ...
2
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1answer
48 views

right Sylow statement?

I would like to know if this statement (which i just met and suspiciously never realized before) and its proof are true: Let $p$, $q$ be distinct primes and $G$ a group of order ...
2
votes
1answer
132 views

counterexample: degree of representation $\leq$ index of normal subgroup

if I have a finite group $G$ with an abelian normal subgroup $N$ and an irreducible representation $\pi$ of $G$ over $K$. Then I know, that $deg(\pi) \leq [G:N]$, if $K$ has positive characteristic ...
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0answers
53 views

reflection groups over finite fields and coxeter groups

Coxeter groups include groups like E6, G2 etc which when defined over finite fields are simple finite groups. Are there coxeter representation for such simple finite groups (like E6(q))?
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0answers
117 views

Rational cohomology of quotient by group action

Let $X$ be a topological space with a continuous action by a finite group $G$. Hopefully under some assumptions on $X$ one can identify the rational cohomology of $X/G$ with the $G$-invariants of the ...
2
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1answer
218 views

Converting a (signed) permutation to a reduced word

I vaguely know that by looking at the inversions of a permutation, you can write down the reduced word expressing the permutation as a product of adjacent transpositions $s_i = (i,i+1)$. However, I ...
2
votes
1answer
254 views

Definition of $K$-conjugacy classes

I would like to understand the definition of "$K$-conjugacy classes" I found in an article by G. Pazderski Pazderski, Gerhard. "On the number of irreducible representations of a finite group." ...
1
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2answers
146 views

Cycling through powers of a generator of finite field. Something similar to modpow for Z/nZ

I am not sure if I am talking to the correct community (perhaps stack overflow is best). I want to be able to compute $$g^x = \underbrace{g \cdot g \cdot \dots \cdot g}_\text{x amount of times}$$ ...
0
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2answers
109 views

Subgroups written as products

Suppose a finite group $G$ is the product of two of its proper subgroups $G=AB$. Assume also that $A\lhd G$ and that $A,B$ have relatively prime orders. Isn't it true that any subgroup $H$ of $G$ can ...
2
votes
1answer
128 views

Normalizer of regular action made linear

For a finite group $G$ the regular action $\rho$ of $G$ on itself (by right multiplication) has the property that the normalizer of $\rho(G)$ in the symmetric group $S_G$ is isomorphic to the ...
2
votes
3answers
142 views

number of automorphism on $\mathbb{Z}_9\times \mathbb{Z}_{16}$

we have to find number of automorphism on $\mathbb{Z}_9\times \mathbb{Z}_{16}$ I know a result which says $Aut(\mathbb{Z}_n)\cong U_n$ where $U_n$ is the multiplicative group i.e ...
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1answer
162 views

The number of cyclic subgroup

Let $p$ be prime divisor of order of finite group $G$, and the number of cyclic subgroup of order $p$ be $p+1$. If $P$ is a Sylow $p$-subgroup of $G$, then $P$ is normal in $G$ and $|P|=p^{2}$($P$ is ...
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3answers
185 views

An example of finite groups

Is there any example of finite group $G$ with the following properties? 1) There is prime divisor $p$ of order $G$ such that the number of cyclic subgroup of order $p$ is $p+1$. 2) The order of ...
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2answers
664 views

Non-isomorphic abelian groups of order $19^5$

I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
1
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1answer
126 views

How to prove $\frac{G}{Z(G)}\cong \frac{\mathbb{Z}}{p\mathbb{Z}}\times \frac{\mathbb{Z}}{p\mathbb{Z}} $

Let $G$ be a group non-abelian group of order $p^3$, where $p$ is a prime number, prove that: $\fbox{1}$ $|Z(G)|=p$ $\fbox{2}$ $Z(G)=G'$ $\fbox{3}$ $\frac{G}{Z(G)}\cong ...
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3answers
873 views

Any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.

$\fbox{1}$ Prove that any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic. $\fbox{2}$ Prove that $\mathbb{Q}$ is not isomorphic to $\mathbb{Q}\times \mathbb{Q}$. Any hints would be ...
10
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0answers
154 views

Official name(s) for a certain type of p-group

I'm implementing a class of groups into Sage (sagemath.org), a computer algebra system, and I'm wondering if it has any official names. I found it in Gorenstein's "Finite Groups." It is there called ...
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1answer
132 views

Question about a question (irreducible representations of a semidirect product)

In the question Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$ the author is talking about irreducible representations of a semi-direct ...
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1answer
141 views

Semidirect product with a $p$ group

let $P$ be a finite $p$-group that acts on a finite group $G$ and assume that $P$ is maximal subgroup of $G \rtimes P$. Show that $G$ is an abelian $q$-group for some prime $q$. Hint: Show that $P$ ...
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3answers
440 views

How many elements of a given order in a finite group

Let $G$ be a finite group and $n_k$ the number of elements of order $k$ in $G$. Show that $n_3$ is even and $o(G) - n_2$ is odd. By Lagrange's Theorem, if $k$ does not divide $o(G)$, there are no ...
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2answers
147 views

Finite non-abelian $p$-group cannot split over its center

Show that a finite non-abelian $p$-group cannot split over its center. I'd be happy for some clues.
2
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2answers
215 views

Finite p-group with a cyclic frattini subgroup.

I have a question about the following theorem that I found in some research. Is it possible that $E$ is the identity? I just found this elaborated proof that might help.
2
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0answers
77 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
2
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2answers
180 views

no simple group of order 945

I need to show that there are no simple groups of order 945. I've tried the regular method using the Sylow theorems. $$|G|=945=3^3\cdot5\cdot7 $$ If $G$ is simple then there should be 7 Sylow-3 ...
3
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5answers
732 views

If $H\unlhd G$ with $(|H|,[G:H])=1$ then $H$ is the unique such subgroup in $G$.

Here is a problem from "An introduction to the Theory of Groups" by J.J.Rotman: Let $G$ be a finite group, and let $H$ be a normal subgroup with $(|H|,[G:H])=1$. Prove that $H$ is the unique such ...