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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-1
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1answer
235 views

Alternative Group Theory Approach

I am trying to learn basic group theory and I have a basic question. Groups are defined in terms of a binary operator and and elements along with their inverses. In category theory, Grp denotes this ...
2
votes
1answer
2k views

Cayley table help

I have a Cayley table with four elements and a binary structure $*$. I know that if I have the same element along the main diagonal (from top right corner to bottom left corner), then the set is ...
2
votes
1answer
156 views

Why is $\mathbb{F}_q^{*}/(\mathbb{F}_q^{*})^n\rightarrow \boldsymbol{\mu}_n: \overline{x}\mapsto x^{(q-1)/n} $ a group isomorphism?

Let $\mathbb{F}_q$ denote a finite field. Let $n\geq 1$ be an integer such that $n\mid q-1$. Hence the $n$-th roots of unity $\boldsymbol{\mu}_n$ are contained in $\mathbb{F}_q$. Why is the map ...
5
votes
1answer
255 views

Why is GL(4,2) isomorphic to Alt(8)?

I am trying to prove that $\operatorname{GL}(4,2)\cong \operatorname{Alt}(8)$. As part of the proof I already know that $\operatorname{Alt}(7)\subset \operatorname{GL}(4,2)$ and that ...
3
votes
1answer
210 views

Divisibility and Sylow p-subgroups

Let $p$ be a prime. Suppose $N$ is a normal subgroup of a finite group $G$. If $n_{p,G}$ is the number of Sylow $p$-subgroups of $G$, and $n_{p,N}$ is the number of Sylow $p$-subgroups of $N$, then ...
3
votes
2answers
184 views

Is $A_5\times A_5$ the only nontrivial normal subgroup of $(A_5 \times A_5) \rtimes C_2$?

The only nontrivial normal subgroups of $A_5 \times A_5$ are $A_5 \times 1$ and $1\times A_5$. What are the normal subgroups of $(A_5 \times A_5) \rtimes C_2$? Is $A_5 \times A_5$ the only ...
3
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1answer
337 views

Is there a non-abelian group of order 49?

Is there any non-abelian group of order $n=49$? I assume there should be at least one but I cannot find an example.
4
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0answers
148 views

A problem of J.D. Dixon

Referring to his paper from 2004, I was wondering if anyone is aware of any relevant work done on the following problem: Of course, the case $w(X_1,X_2)=X_1X_2X_1^{-1}X_2^{-1}$ admits the answer ...
1
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1answer
257 views

left and right coset verifications

How to find the left and right cosets of the subgroup $H = \{r_0, s_0\}$ of $D_4$? And are they the same? If we let $H' = \{r_0,r_2\}$, are the left and right cosets the same, where $H'$ is a ...
4
votes
1answer
579 views

Subgroup of maximal order is normal

I just did this problem: "Prove that if $G$ is a finite group and $H$ is a proper normal subgroup of largest order, then $G/H$ is simple." And I am currently working on this problem: "Suppose that ...
11
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1answer
225 views

Class equation of subgroup of $SL(4,\mathbb{F}_2)$

Can you point me toward a computation-light derivation of the class equation of the subgroup of $SL(4,\mathbb{F}_2)$ consisting of upper-triangular matrices with 1's on the main diagonal? The ...
3
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2answers
461 views

Structure theorem for finitely generated abelian groups

How can we use fundamental theorem of finitely generated abelian groups to list all abelian groups of order 16 up to isomorphism.
5
votes
1answer
165 views

How do I write a group action in terms of the action of a subgroup?

Suppose $H < K < G$ are finite groups and $G$ acts primitively by (right) multiplication on the set $\Gamma = G/K$ of (right) cosets of $K$ in $G$, and $K$ acts primitively on the set $\Delta = ...
3
votes
1answer
216 views

Proving a theorem about a finite simple group

I need to prove this theorem that I had in an exam and I am stuck. Let $G$ be a finite simple group and we assume that for every prime $p$ the number of $p$-sylow sub-groups is $\leq 6$. Prove ...
2
votes
1answer
263 views

The number of p-elements in a finite group

Let $p$ be a prime number. Can one find always a finite non-$p$-group $G$, such that the portion of number of $p$-elements of $G$ comes arbitrarily close to the total number of elements of $G$. That ...
7
votes
2answers
311 views

A condition for a finite group to be abelian

I know that a group of order $pq$ where $p < q$ and $p \not\mid q-1$ is abelian (in particular it is cyclic), and that a group of order $p^2q^2$ where $p$ and $q$ are distinct primes, $q \not\mid ...
6
votes
1answer
535 views

How many homomorphisms are there from $\mathbb{Z}_{n}$ into $S_{n}$?

I would like to know how many homomorphisms there are from $\mathbb{Z}_{n}$ into $S_{n}$? If $n=2$ or $n$ is odd, I think that there are $(n-1)!+1$. I am counting those cycles of order $n$, when $n$ ...
16
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5answers
577 views

Upper bounds on the size of $\operatorname{Aut}(G)$

Any automorphism of a group $G$ is a bijection that fixes the identity, so an easy upper bound for the size of $\operatorname{Aut}(G)$ for a finite group $G$ is given by ...
7
votes
3answers
701 views

Explicit descriptions of groups of order 45

I know that there are two groups of order 45, and obviously one of them (up to isomorphism) is $\mathbb{Z}_{45}$. I'm trying to understand explicitly what the structure of the other is like. By ...
3
votes
3answers
340 views

subgroup of finitely generated solvable group is finitely generated (false proof)

Can't find a flaw in that proof: Induction by the length of derived series. Base: if $[G, G]=e$ then the group is abelian... Assume that statement is true for n-1. We have group $G$ with the ...
13
votes
2answers
686 views

Has this “generalized semidirect product” been studied?

If $G$ is a finite group with subgroups $H$ and $K$ such that $HK = G$ and $H\cap K = \{1\}$ we get that every element of $G$ can be written uniquely as $hk$ with $h\in H$ and $k\in K$. This then ...
7
votes
1answer
283 views

how to prove that semidirect products are not isomorphic

For example, I want to understand what are different $S_5 \rtimes \langle c\rangle_2$ products. $\mathrm{Aut}(S_5)=\mathrm{Inn}(S_5)\simeq S_5$, so we can have direct product or $\psi: с \rightarrow ...
2
votes
1answer
261 views

Isomorphism between a quotient group and the 2-torsion subgroup

Let $G$ be a finite abelian group, and let $2G$ denote the subgroup $\{ g * g : g \in G\}$. Let $G[2]$ be the 2-torsion subgroup of $G$. I want to show that $$ G/2G \cong G[2]. \qquad (1) $$ The ...
4
votes
2answers
199 views

An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group

Exercise $7$, page 51 from Hungerford's book Algebra. Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and ...
3
votes
3answers
256 views

Why aren't $C_2 \times C_3$ and $D_6$ isomorphic?

I am new to group theory so please forgive me for my stupid question. Why are the groups $C_2\times C_3$ not the same as $D_6$? Aren't they both generated by 2 elements one of order 2 the other 3? Is ...
7
votes
2answers
1k views

Every Transitive Permutation Group Has a Fixed Point Free Element

If $G$ acts transitively by permutations on a finite set $A$ with more than one element (i.e. $G$ is a transitive permutation subgroup of the symmetric group $S_A$). Why does $G$ necessarily contain ...
1
vote
1answer
126 views

How many nonisomorphic normal subgroups does $S_{n}$ have?

I am self-studying Group Theory. I know that $A_{n}$ is a normal subgroup of $S_{n}$, but I've realized that I don't know another one, I mean, another subgroup of $S_{n}$ that is a normal subgroup of ...
1
vote
1answer
121 views

A generator for $(\mathbb{Z}/p^k\mathbb{Z})^{\times}$

Let $a$ be the generator for the group of units $\bmod{p^k}$ for $p$ an odd prime and k a positive integer, i.e. $\langle a \rangle= (\mathbb{Z}/p^k\mathbb{Z})^{\times}$. Is it true that $a$ is also a ...
2
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2answers
229 views

Constructing a subgroup of a given order

Given a finite group $G$ and a positive integer $k$ that divides the order of $G$, is there some sort of algorithm or other systematic method for constructing a subgroup of $G$ of order $k$? In ...
1
vote
1answer
97 views

Endomorphisms of $\mathbb{Z}/6\mathbb{Z}$

I have a question, which may be trivial, but I really don't understand how I should solve the next exercise: List all group endomorphisms of $\mathbb{Z}_6$. I consider that I must write down ...
3
votes
1answer
172 views

If $H \unlhd G$ is CC-closed, then $H$ is a Hall subgroup

I'm stuck with this problem. Let $H$ be a normal subgroup of a finite group $G$ such that $C_G(x)\subseteq H$ for every non-identity element $x\in H$ (that is, $H$ is a normal CC-subgroup of $G$). ...
2
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0answers
139 views

Are there 16 or 24 automorphisms of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$?

In this question I said that the automorphism group of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ has 16 elements because If $\varphi$ is one of this automorphism then ...
0
votes
1answer
224 views

Induced representation of symmetric group.

Im stuck with this one and I don't even know how to start, I would appreciate any help: Can you describe the induced representation of the standard representation of $S_{n}$ in $S_{n+1}$?
4
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0answers
132 views

Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a finite group $G$?

I know that if $H$ and $K$ are subgroups of a finite group $G$, then $|HK|=\frac{|H||K|}{|H\cap K|}$. Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a ...
6
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1answer
310 views

If $S$ and $T$ are nonempty subsets of a finite group $G$, then either $G=ST$ or $|G|\geq |S|+|T|$.

Exercise 2.29 from Rotman's book An Introduction to the Theory of Groups. (H.B.Mann) Let $G$ be a finite group, and let $S$ and $T$ be (not necessarily distinct) nonempty subsets. Prove that ...
2
votes
1answer
165 views

Simple proof for finite groups that $g^{\#(G)}=1$ [duplicate]

Possible Duplicate: Is Lagrange's theorem the most basic result in finite group theory? I can't seem to find a simple proof of this in my textbook, not can I figure out a good way to ...
2
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1answer
555 views

Symmetric group and commuting elements

What is the normalizer of the subgroup $\langle(1,2,...,p)\rangle$ in $S_p$? Clearly it will have to contain the subgroup $\langle(1,2,...,p)\rangle$, but also some additional $p-1$ elements that ...
4
votes
1answer
154 views

How do I create a group action table with GAP?

Background: Let $G$ be a group of size $k\cdot p^n$. Let $S$ be the set of all subsets of size $p^n$ of $G$. Define the map $f\colon G \times S \rightarrow S$ by $(g, s) \mapsto gs$ if $s \in S$. ...
2
votes
1answer
192 views

$HK$ is not a subgroup

This is Exercise 7, page 40 from Hungerford's book Algebra. Let $G$ be a group of order $p^{k}m,$ with $p$ prime and $(p,m)=1.$ Let $H$ be a subgroup of order $p^{k}$ and $K$ a subgroup of order ...
4
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1answer
974 views

Alternative way to prove that a group of order 160 is not simple

Is there a neat way to show that a group of order 160 is not simple without directly quoting Poincare's theorem? I am thinking of maybe using the Sylow theorems to say that in order that the ...
5
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0answers
209 views

Icosahedral symmetry as permutation group

Hopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the ...
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0answers
57 views

Inner tensor and restriction

Let $\pi$ resp. $\chi$ be a finite dimensional representation resp. 1-diml. rep. of a finite group. We define $\chi \otimes \pi (g) = \chi(g) \pi(g)$ as a rep of $G$. For $N$ subgroup, does hold ...
10
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1answer
352 views

How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?

How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$? If $\varphi$ is one of this automorphism then $\varphi((1,0))=(1,0),(1,3),(3,3),(3,0)$ and ...
4
votes
1answer
126 views

Why must such a group be dihedral?

I tend to think dihedral groups are easy to recognize, but I don't quite see why if G is a quotient of $$U = \langle x, y, z : x^2 = y^2 = z^2 = 1, yx=xy, zy=yz \rangle$$ and G has order 4 mod 8 (so, ...
3
votes
2answers
459 views

Generator of cyclic groups

I have a group of order $13\times 11\times 7$. I am able to show that my group is abelian (using a combination of Sylow's theorems and seeing that the intersection of the 3 cyclic subgroups ...
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2answers
247 views

Solvability and Simplicity

I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. So I looked up Burnside's theorem and saw that it doesn't mention "simple" explicitly, rather it says ...
3
votes
1answer
110 views

Given an $n$, are there infinitely many finite fields $F$ such that none of the orders of $PSL_n(F)$ divide each other?

Given an integer $n$, is there an infinite set of finite fields $F_i$, $i\in \mathbb{N}$ such that for $i\neq j$ we have that $|PSL_n(F_i)|$ does not divide $|PSL_n(F_j)|$. The motivation is that ...
2
votes
2answers
215 views

A problem concerning action via automorphisms

This problem asks to show that if $A$ acts on $G$ via automorphisms, where either $A$ or $G$ is soluble and both $A$ and $G$ are finite groups and $G$ is nontrivial, then $G$ possesses an ...
2
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2answers
161 views

$A_5$ problem with normal subgroup

Let G=$A_5$ and $H=\bigl\langle (12)(34),(13)(24)\bigr\rangle$. Prove $(123) \in N_{G}(H)$ and hence deduce the order of $N_{G}(H)$. I know you claim that $A_5$ is simple, then $N_{G}(H)$ has ...
5
votes
1answer
426 views

Proof of no simple group of order 992

Prove there are no simple groups of order $992$. Factorise it. $31 \times 2^5 $ so you have $|G|=31 \times 2^5 \geq n_{31}(31-1)+ n_{2}(2^5-1)+1$ Putting it in Sylow theorem. So how do you get ...