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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5
votes
2answers
102 views

If $p\mid|G|$ then how many elements of order $p$ are there in $G$?

Let $G$ be a finite group and $p$ be a prime such that $p\mid|G|$ , then obviously $G$ has an element of order $p$ (by Cauchy's theorem) ; I would like to know exactly how many elements of order $p$ ...
0
votes
1answer
48 views

The comultiplication on $\mathbb{C} Q $ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
0
votes
1answer
115 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
1
vote
6answers
799 views

Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
11
votes
0answers
194 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = \sup\{\...
3
votes
1answer
208 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
2
votes
1answer
119 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
2
votes
0answers
68 views

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $$|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$$ for each b fixed & for all $a+b$ such that $p^{a+b}...
1
vote
2answers
49 views

Does the following binary operation form a group on a set with 3 elements? (multiple identities?)

Let S = {a, b, c}. *| a b c ----------- a| a b c b| b a a c| c a a This seems to have all the desired characteristics of a group, however, both b and c ...
4
votes
1answer
80 views

How to determine the isomorphism types of given groups with generators and relations

I was classifying the all groups of order 30 and I got the following groups $\langle a,b \mid b^{-1}ab=a^4, a^{15}=b^{2}=1\rangle$ and $\langle a,b \mid b^{-1}ab=a^{11}, a^{15}=b^{2}=1\rangle$. How ...
1
vote
0answers
39 views

How to reconstruct geometric object that a Frobenius group acts on

A Frobenius group has equivalent definitions: a transitive permutation group on a finite set such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. ...
2
votes
1answer
70 views

$GL_2(R)$ and $\mathbb{A}_4$

So, i have to prove that $\mathbb{A}_4$ is not isomorphic to a subgroup $G$ of $GL_2(R)$. Here is a "hint" (are previuos part of the same problem that i was thinking, so i supose should help): If $H$...
2
votes
1answer
130 views

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
0
votes
2answers
48 views

Show that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+) $ holds

I want to show, that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+)$ holds with $p$ prime. ($\mathbb{F}_p^+$ is the additive group, $\mathbb{F}_p^\times$ multiplicative group) As hint we ...
1
vote
1answer
56 views

Proof of the theorem that relates the size of the conjugacy class to the order of the centralizer subgroup

Could you please explain the step in this theorem and proof that I do not follow: If $G$ is a finite group, let $C_{G}(x)$ be the centralizer of $x$ in $G$, that is $C_{G}(x)=\{g \in G: xg=gx\}$ ...
3
votes
3answers
302 views

Motivation and intuition of double cosets

In Dummit and Foote, there was a problem on double cosets. Let $H$ and $K$ be subgroups of $G$ and $x\in G$, $HxK$ is the double coset of $x$. I can prove the double cosets partitions $G$. However, I ...
2
votes
2answers
137 views

A Gap code for the alternating group $A_4$

I need a GAP code for checking the following question: Is it true that for every subset $A$ of the alternating group $A_4$ with $4$ elements there exists a subset $B$ of order $3$ such that $A_4=AB$?...
0
votes
2answers
114 views

Use Sylow's theorem to show that $G = HN_G(P)$

I am stumped on this question. Does anyone have some helpful hints or a solution to this question? Thanks! Let $G$ be a group of finite order. Let $H$ be a normal subgroup of $G.$ Let $P$ be a ...
4
votes
2answers
593 views

Normalizer and centralizer of abelian subgroups of a group are equal [closed]

I have a question: Let $G$ be a finite group. If for each abelian subgroup $H$ of $G$ the centralizer and the normalizer of $H$ are equal, that is, $C_G(H)=N_G(H)$, prove that $G$ is abelian group....
10
votes
3answers
111 views

Group of order $224$

Any one can give a hint to prove this? Every group of order $224$ have a subgroup of order $28$.
5
votes
2answers
122 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
1
vote
1answer
57 views

Frobenius dihedral groups

How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?
2
votes
1answer
129 views

Characteristically simple subgroup

Let $G$ be a finite group and $H$ be a minimal normal subgroup of $G$. How can we show that $H$ is a characteristically simple subgroup of $G$?
2
votes
1answer
39 views

List of groups with specific divisors

I want to find list of finite simple nonabelian groups which their orders divisor lies in specific set of primes for example the set $\{2,3,5,7,11\}$. Is there a method to do this? Would Zsigmondy's ...
3
votes
1answer
110 views

Minimal normal subgroup in a finite group

Let $G$ be a finite group and $G=HK$ such that $H<G$. How can we show that if $K$ is an abelian minimal normal subgroup of $G$, then $H$ is a maximal subgroup of $G$ and $H\cap K=1$?
3
votes
2answers
120 views

Elements of finite groups have finite order.

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
5
votes
1answer
706 views

Is there any efficient algorithm for finding subgroups of a given finite group?

I am implementing an algorithm which finds every subgroup of given group. Here's my algorithm. Let $G$ be a group of order $n$ with elements $g_1,\cdots,g_n$. Then I consider each $\langle g_i\...
2
votes
2answers
89 views

Cardinality of Centralizer of some element

Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?
4
votes
2answers
57 views

A question on terminology (Group Theory)

Is there a standard adjective to describe a finite group $G$ of composite order which possesses, for each (positive) divisor $d$ of $|G|$, a subgroup of order $d$? I would guess "Lagrangian" but I ...
1
vote
4answers
67 views

A finite group of order $n$, having a subgroup of order $k$ for each divisor $k$ of $n$, is not simple?

I was asked to prove that, if a finite group $G$ of order $n$ has a subgroup of order $k$ for each divisor $k$ of $n$, then $G$ is not simple. I tried to do this but I could not. Can anyone please ...
6
votes
1answer
212 views

Why do we care about simple groups rather than indecomposable groups?

A first thing that we do to analyze something is "divide and conqur." So it is natural to consider simple groups. In the same way, it appears to be natural to consider indecomposable groups. However, ...
4
votes
1answer
99 views

On finite 2-groups that whose center is not cyclic

Let $G$ be a finite 2-group such that $\left|\dfrac{G}{Z(G)}\right|=4$, $Z(G)$ is not cyclic and $Z(G)$ has at least one element of order 4. Then prove that there exists an automorphism $\alpha$ of $...
0
votes
1answer
31 views

Equality of powers of elements in a subset of a generating set for a finite group.

I'm currently reading a paper on finding Hamiltonian paths in Cayley graphs and the author makes a claim that I can't seem to understand. Let $G$ be a finite, nilpotent group, $N$ a normal subgroup ...
5
votes
1answer
225 views

Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

An exercise from Dummit and Foote pg. $101$ ex. $9$ asks to show the following: Let $G$ be a group of order $p^{a}n$ where $p$ does not divide $n$ and let $N\unlhd G$ so that $|N|=p^{b}m$ where $...
3
votes
2answers
132 views

A finite divisible group is trivial

I am having trouble seeing why a finite divisible group is necessarily trivial. Why does this have to be the case?
1
vote
1answer
40 views

Decomposing $g=xy$ where $\left|g\right|=\left|x\right|{\cdot}\left|y\right|$

Throughout, let us assume we are working with a finite group $G$. The order of an element $g\in G$ is denoted by $\left|g\right|$. It is a standard exercise to prove that if $x, y\in G$ have ...
2
votes
0answers
41 views

Order of a group of matrices

I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$. So I was thinking on the homomorphism $\det: G \rightarrow \mathbb{Z}...
1
vote
0answers
38 views

A question regarding conjugacy classes of central involutions.

An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$. Clearly if an involution is central then its ever conjugate is also ...
2
votes
2answers
97 views

Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
1
vote
1answer
93 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that $...
9
votes
2answers
111 views

$A_n \times \mathbb{Z} /2 \mathbb{Z} \ \ncong \ S_n$ for $n \geq 3$

I try to show that $A_n \times \mathbb{Z} /2 \mathbb{Z} \ \ncong \ S_n$ for $n \geq 3$. It is not hard to show the statement for $n=3$. We have $$ A_3 \times \mathbb{Z} /2 \mathbb{Z} \ \cong \ ...
2
votes
2answers
60 views

Show that there is just one subgroup $H \subset S_4$ such that $[S_4:H] = 2$

I had to show that $S_4$ has one subgroup of index $2$. Below, you'll find what I tried to do so Of course, $A_4$ is a subgroup index $2$. To show that there is not another subgroup with the same ...
4
votes
2answers
263 views

Subgroups of groups of order 36

Is there any group of order 36 with no subgroup of order 6? Is there any group of order $p^2q^2$ with no subgroup of order $pq$? Is there any group of order $p^{2m}q^{2m}$ with no subgroup of order $p^...
4
votes
1answer
107 views

Unique normal subgroups of every possible order

Can one characterize groups $G$ for which there is a unique normal subgroup of order $d$ for every divisor of the order of $G$? For example must they be solvable?
5
votes
5answers
267 views

If a group contains a subgroup with the order of each of its divisors, is it abelian?

Let $G$ be a group that has a subgroup of size $d$ for every $d$ that divides $|G|$. Must $G$ be abelian? It can be shown using complete induction that the converse of the above statement is true, ...
3
votes
1answer
95 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order $...
0
votes
1answer
109 views

Find a subgroup of $S_4$ that is isomorphic to V, the Klein group.

So I know that the Klein group is the group with 4 elements that is not cyclic but I'm stuck from there onwards?
1
vote
1answer
84 views

The number of subgroups conjugate to a given subgroup of a finite group

Let $H$ and $K$ are subgroups of $G$ conjugate to each other. $A$ is defined as $$A = \{a \in G \mid aHa^{-1} = H \}$$ for all $a\in G$. Prove that $A$ is a subgroup of $G$ and prove that if $G$ ...
0
votes
1answer
82 views

2 question on solvable group's property

A theorem says a finite group G is solvable if and only if for every divisor n of $\vert G\vert$ such that (n,$\frac{\vert G\vert}{n})=1$,G has a subgroup of order n. Does this imply that G must ...
1
vote
0answers
76 views

how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...