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
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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 ...
0
votes
2answers
110 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
562 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 ...
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
120 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
54 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
106 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
118 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
657 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 ...
2
votes
2answers
84 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
66 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
210 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
98 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 ...
4
votes
1answer
214 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
131 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 ...
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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
90 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
254 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 ...
4
votes
1answer
105 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
265 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
94 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
108 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
74 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 ...
1
vote
2answers
258 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
3
votes
0answers
52 views

Groups of order $n^2$ with no subgroup of order $n$ [duplicate]

Is it possible to classify those groups whose order is $n^2$ for some natural number $n$ but which do not have any subgroups of order $n$? To be a bit more specific (in case a full classification is ...
9
votes
2answers
230 views

Groups of order $n^2$ that have no subgroup of order $n$

For which $n$ is there a group of order $n^2$ without a subgroup of order $n$. Such groups can not be nilpotent. This question is related to Sudokus as composition tables of finite groups.
0
votes
1answer
92 views

N is a normal subgroup of G if $aNa^{-1} \subset N $ for all $a ∈ G$. Prove that in that case, $aNa^{-1} = N $.

I said if N is a normal subgroup of G when $aNa^{-1} \subset N $ aN = Na as N is a normal subgroup of $G$. Therefore $aNa^{-1} = Naa^{-1} $ and $aNa^{-1} = N $. I would like to go with this proof ...
2
votes
1answer
105 views

Structure of a group, $G$, of order $pq$ where $p, q$ are prime.

There is a proposition in Beachy and Blair's Abstract Algebra that I don't entirely follow. The proposition is the following: Let $G$ be a group of order $pq$, where $p > q$ are primes. a) If ...
1
vote
0answers
66 views

Counting apartments in spherical buildings

Is there a formula for the number of apartments in a finite, spherical building? To be specific, is there a formula that depends on the associated Coxeter group and the thickness of the building? ...
3
votes
1answer
120 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
1
vote
4answers
398 views

If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$

I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms. So, is the ...
2
votes
1answer
77 views

Relation of order of a permutation with its sign

Let $G$ be a group with order $2m$ where $m$ is odd. Consider the left action $\lambda_g:G\to G$. It appears that if $g$ has odd order iff $\lambda_g$ has odd order iff $\lambda_g$ is an even ...
0
votes
1answer
42 views

pemutation representation that confuses me a lot recently

For any group G, define group action on a set A. There will be a permutation representation of that group action. I am kind of confused why the permutation representation can be used to reflect the ...
3
votes
1answer
91 views

Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
1
vote
2answers
71 views

Normal subgroup created by a bunch of elements

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather ...
1
vote
1answer
91 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
1
vote
1answer
45 views

Is my proof correct? (about commutators)

I want to prove the following fact: Let $G$ be a finite group and let $x, y\in G$ such that $[x, y] \in Z(G)$. Then $[x, y^s] = [x, y]^s$ for every $s\in \mathbb{Z}$. If we assume that $[x^r, y^s] ...
4
votes
1answer
74 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
2
votes
2answers
65 views

Determine the center of this finitely presented group.

Consider the group $G(n)= \langle a, b \ \vert\ aba^{-1}=b^{n+1}, bab^{-1}=a^{n+1} \rangle$, $n\ge 1.$ Show that the center $Z$ is cyclic of order $n$ and that $G/Z$ is abelian of order $n^2$. This ...