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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1answer
389 views

Find the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n

What is the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n? This is a question in Joseph A.Gallian's book, and the answer is n=36 and the ...
1
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0answers
59 views

Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
4
votes
1answer
61 views

Can we always construct a “$p$th root” of a $p$-element in a finite group?

Let $p$ be a prime, $G$ a finite group, and $g\in G$ a $p$-element. Can one always embed $G$ in a finite group $H$ that contains a $p$-element $h$ such that $h^p=g$?
1
vote
1answer
39 views

Obtaining a presentation of the dihedral group from a semidirect product

I am working on classifying groups of order 44. I have shown that $G\cong P_{11} \rtimes_{\varphi} P_{2}$, where $P_p$ are Sylow p-subgroups and $\varphi:P_{2} \to $Aut$(P_{11})$ is a group ...
0
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1answer
77 views

if $G$ is finite group then polycyclic group is equivalent to super solvable group?

I don't know why this is true? can you help me: if $G$ is finite group then polycyclic group is equivalent to super solvable group Definitions- Polycyclic group $G$ is a polycyclic if has a ...
1
vote
1answer
60 views

Colouring a tetrahedron

How would I write down the elements sr and $sr^2$ of G as a product of disjoint cycles? If I am looking for the orbits of this action, do I have 4 orbits $\{1\:2\:3\}, \{p12\:p23 ...
3
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2answers
66 views

$G$ a finite group, $H$ a subgroup of index $2$ in $G$. If $K$ a subgroup of $G$ of odd order then $K$ contained in H.

Let $G$ be a finite group and $H$ a subgroup of $G$ such that $|G:H|=2$. Suppose $K$ a subgroup of $G$ of odd order. Show $K$ is contained in $H$. I'm stuck. Need a hint.
5
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2answers
67 views

Classification of all finite elementary $p$-groups.

Let $G$ be a finite group. For a prime number $p$, let us call $G$ an elementary $p$-group iff $\exp G=p$. I know that all elementary $2$-groups are abelian, and I also know the construction of ...
4
votes
0answers
52 views

Is a finite group which is generated by two fully invariant abelian subgroups always abelian?

Let $G$ be a finite group satisfying there exist two fully invariant subgroups $H,K$ of $G$ such that $H$ and $K$ are abelian and generate the whole group $G$. Then can we conclude that $G$ is ...
1
vote
1answer
52 views

under following conditions $G$ has only one subgroup of order $p$.

Let $|G|=p^m$ for $m \ge 2$. If every subgroup of $G$ of order $p^2$ is cyclic, then $G$ has only one subgroup of order $p$.
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0answers
39 views

Generaliation of semidirect product

Let $G=\langle X\mid R\rangle$ and $H=\langle Y\mid S\rangle$, and let $\phi :H \to \textrm{Aut}(G)$. Then the semidirect product $G\rtimes_{\phi}H$ has the following presentation: $G\rtimes_\phi H = ...
1
vote
1answer
36 views

Finding subgroups of $D_{12}$

Let $G=D_{12}=\langle a,b \mid a^6=b^2=e, bab^{-1}=a^{-1} \rangle$. Find all subgroups of $G$. We can easily spot the cyclic normal subgroup $C=\langle a \rangle = \{e,a,a^2,\dots,a^5\}$. Now ...
0
votes
2answers
41 views

homomorphism between $K = \langle u,v \mid u^2 = 1, v^4 =1, uv=vu\rangle$ and $ \mathbb{Z}_4$

Let $K = \langle u,v \mid u^2 = 1, v^4 =1, uv=vu\rangle$ be a group. Is there a homomorphism between K and $ \mathbb{Z}_4$?
1
vote
4answers
187 views

Order and index of a normal subgroup $N$ are relatively prime

Let $N$ be a normal subgroup of a finite group $G$. Assume that the order of $N$ and the index of $N$ in $G$ are relatively prime. Prove that if $g\in G$ satisfies $o(g)\mid o(N)$, then $g\in N$. ...
7
votes
1answer
70 views

If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple

I'm looking for verification: My claim: If $G$ is a finite group and $H$ is a (proper)subgroup of index $k>1$, where $k! \leq |G|$, then $G$ is not simple. Proof: Consider the set of left cosets ...
2
votes
1answer
50 views

Disjoint normal subgroups - one contained in the centralizer of the other

Let N and M be normal subgroups of a group G and assume that N and M have only one element in common. Prove that N is contained in $C_G(M)$. First I concluded that |NM|=|N|*|M|. Now I'm trying to ...
1
vote
2answers
91 views

Proving a quotient group is not Abelian without calculating actual cosets

Given the normal subgroup of S4: N={(1),(12)(34),(13)(24),(14)(23)}, show that S4/N is not Abelian. What I did was to calculate two random cosets of N in S4,like in the picture I attached, and show ...
2
votes
1answer
73 views

$H$ and $K$ are subgroups of $G$.Recall that $HK=\{hk:H\in H,k\in K\}$ Show that $|HK|=|H||K|/|H\cap K|$

Let $G$ be a finite group, and let $H$ and $K$ be subgroups of $G$.Recall that $HK=\{hk:H\in H,k\in K\}$ Show that $|HK|=|H||K|/|H\cap K|$
2
votes
3answers
81 views

Finding the order of Z(G) in a non-Abelian group of order 8 [duplicate]

Let $G$ be a non-Abelian group of order $8$. Prove that $|Z(G)|$ is less or equal to $2$. First I must say this is a question about normal subgroups. I haven't yet studied homomorphisms or more ...
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3answers
110 views

The center of a non-Abelian group of order 8

Let G be a non-Abelian group of order 8. Prove that $|Z(G)|\leq2$. (The center $Z(G)$ is defined as $Z(G)=\{ a\in G | ag=ga$ for all $g\in G \}$). I deduced from Lagrange's theorem that ...
3
votes
1answer
102 views

Inducing representation for groups of order $p^3$

For groups $G$ such that $|G|=p^3$ one can show that $(1)$ $Z=Z(G)\cong C_p$ $(2)$ $G'=Z$ $(3)$ $G/Z \cong C_p \times C_p$ Take any $x \in G/Z$. Then $N=\langle x,Z \rangle$ is an abelian normal ...
0
votes
1answer
37 views

Properties of groups of order $p^3$

For non-abelian groups $G$ such that $|G|=p^3$ one can show that $(1)$ $Z=Z(G)\cong C_p$ $(2)$ $G'=Z$ Now want to show that $G/Z \cong C_p \times C_p$. So, If $G/Z \cong C_{p^2}$, take its ...
3
votes
3answers
45 views

Finding the order of $\mathrm{GL}_n(\mathbb{F}_p)$

Is there a way to find the order of the group $\mathrm{GL}_n(\mathbb{F}_p)$. In my notes for $\mathrm{GL}_3(\mathbb{F}_2)$ it is done by brute force but this does seem like a very good method.
2
votes
0answers
32 views

Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
2
votes
1answer
86 views

Definitions of $\mathrm{Hom}(V,W)$

I have the definition of a homomorphism as map such that $\varphi(g_1g_2)=\varphi(g_1)\varphi(g_2)$ I have the definition of $\mathrm{Hom}(V,W)$ as $$\begin{align}\mathrm{Hom}(V,W) &= ...
0
votes
1answer
33 views

Finding subgroups of $D_{2p}$

Let $p$ be an odd prime Find all the subgroups of $D_{2p}$. We know that all $g^i$ $(i=1,\dots,p-1)$ have order $p$ and all $g^ih$ $(i=0,\dots,p-1)$ has order $2$. By Lagrange if $H < G$ then ...
3
votes
3answers
95 views

Extend isomorphism of subgroups to homomorphism of groups

Given two finite groups $G_1$ ang $G_2$ with respective subgroups $H_1$ and $H_2$ satisfying $H_1\cong H_2$ via the isomorphism $\phi$, is it always possible to extend $\phi$ to an homomorphism ...
0
votes
2answers
134 views

Proof involving Cyclic group, generator and GCD

Theorem: $$\left\langle a^k \right\rangle = \left\langle a^{\gcd(n,k)}\right\rangle$$ Let G be a group and $$ a \in G$$ such that $$|a|=n$$ Then: $$\left\langle a^k \right\rangle = \left\langle ...
8
votes
2answers
220 views

Divisor of a finite group

Suppose we have a finite group $G$ and $d\in \mathbb N$ is a divisor of $|G|$. We define the set $E_d= \{g\in G : g^d =1\}$. Prove that $d$ is also a divisor of $|E_d|$. So far I proved that ...
4
votes
1answer
45 views

The relationship of subnormal subgroups and modular subgroups of a finite group.

Let $G$ be a finite group, a subgroup $H$ of $G$ is called subnormal if it's a term of a composition series of $G$, and is called modular if it's a modular element of the subgroup lattice $L(G)$. My ...
2
votes
1answer
52 views

Groups occuring as derived subgroups.

I want to solve this problem but I have no idea how to start it. If you know please hint me, thanks. Suppose that $G$ is a group that has subgroup which is cyclic, characteristic and not in the ...
0
votes
1answer
48 views

Criterion for $a^i=a^j$ proof

Let G be a group and let a be an element in G. If a has infinite order, then $$a^i=a^j$$ if and only if $i=j$ If $a$ has finite order, say $n$, then $$\left \langle a \right ...
2
votes
1answer
122 views

What is the use and motivation for this particular concept in permutations?

Say you have the permutation $(54231)$ element of $S_5$ Now you drop say the "4" and then re-rank the remnant permutation on the other elements. Then you are left with, $(4231)$ element of $S_4$ ...
2
votes
2answers
69 views

A normal subgroup so that any homomorphism into a $p$-group is trivial on it.

Problem Let $G$ be a finite group of order $n$ and $p|n$. Show that there is a unique normal subgroup $N$ satisfying the following property: $G/N$ is a $p$-group (I guess it can be trivial ...
5
votes
4answers
368 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
1
vote
1answer
49 views

Natural action of $S_n$ on $\{ 1,2,\dots,n \}$

From reading online the "natural" action of $S_n$ on $\{ 1,2,\dots,n \}$ is $(g,x) \mapsto gx$. How is this action transitive? As far as I can see if we take $g$ to fix some element we will not get a ...
1
vote
1answer
48 views

How can you tell if a normal subgroup induces a semidirect product?

Suppose I have some (finite) group $G$ and a normal subgroup $N$. I know there's no full characterization of whether $G \cong N \rtimes G/N$, but are there well-known tests I can use to answer the ...
3
votes
2answers
102 views

Fiber product of groups

Let $f: G \longrightarrow K$, $g: H \longrightarrow K$ be group homomorphisms. For the fiber product $G \times_K H$ to exist, is it necessary to require that $f$ and $g$ be surjective? I can't see ...
0
votes
1answer
74 views

A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
0
votes
2answers
84 views

Abelian group which is not one of these

Im struggling to find a finite abelian (commutative , associative) group $(G,\circ)$ with some specific conditions: $a\circ b$ isn't naive addition $a+b$ for $a,b\in G$ $G$ is a subset of ...
2
votes
3answers
770 views

Verifying if a multiplication table is from a group

I'm asked to verify which of these multiplication tables form a group. I'm having problems to see which of the axioms for a group are violated in each table. In (a), I couldn't find an element $e$ ...
5
votes
0answers
143 views

Aut(G) is abelian

I've heard of this (open?) problem: Classify groups G such that Aut(G) is abelian. What I discovered: Any characteristic abelian subgroup is cyclic. Center is cyclic. Commutators are cyclic. ...
38
votes
3answers
2k views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
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votes
1answer
31 views

Classifying the central product HK of two cyclic groups [closed]

Let group $H$ be a direct product of cyclic groups $C_1$ and $C_2$ of order $p$ and $p^2$ respectively. Let $D=\{x\in H\mid \text{ord}(x)\leq p \}$. D is generated by $C_1$ and subgroup $E$ of $C_2$ ...
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0answers
21 views

Finding efficiently finite groups whose set of commutators is not a subgroup

The set of commutators of a group might not be a subgroup of the group. I give here such an example from P.J. Cassidy. It is an infinite group. It is possible to derive from this example one of a ...
0
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0answers
46 views

I want to know if the below sentence is true and why?

I want to know if the below sentence is true and why? Let $G$ be an insoluble finite group then there exists $\pi\subset\pi(G)$ such that if $K=O_{\pi}(G)$ and $\bar{G}=G/K$ it follows that ...
2
votes
1answer
36 views

The generating set of Cayley graphs over $Z_n$

Say we have a undirected and connected Cayley graph over $\mathbb{Z}_n$, with generating set $S=\{\pm x_0,\pm x_1,\ldots,\pm x_k\}$. Is it true that we can assume without the loss of generality that ...
1
vote
2answers
32 views

How to prove that $a^{|G|}=e$ if a $\in G $

How to prove that; $a^{|G|}=e$ if a $\in G $ if $G$ is a finite group and $e$ is its identity. I think this could be done through pigeonhole principle but I don't want to use the Lagrange ...
0
votes
1answer
61 views

all of subgroups of group

Is the way to gust that in finite group how many subgroup of same order?I ask this question because when draw the lattice diagram of subgroups of group sure that all of them describe. Thanks for hint
1
vote
1answer
94 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...