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
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2answers
178 views

$p$-Group as union of subgroups

It is well known that a group can not be union of two proper subgroups. For finite $p$-groups, we can say more: A finite $p$-group can not be union of $p$ proper subgroups. Moreover, ...
5
votes
2answers
441 views

If every element has prime power order and $Z(G) \neq 1$ then $G$ is a $p$-group.

In a finite group $G$ if every element is of some prime power order (prime may vary with element) and if $G$ has non trivial center then prove that $G$ is actually of prime power order. Deduce that ...
1
vote
1answer
80 views

Euler's formula and subgroups of $\mathbb Z_n$

Prove that in $\mathbb Z/n\mathbb Z$ for every divisor $d$ of $n$ there is a unique subgroup of order $d$ using the following results: $\sum_{d\mid n}\varphi(d)=n$ and the number of generators of ...
2
votes
2answers
102 views

Number of conjugacy classes of the reflection in $D_n$.

Consider the conjugation action of $D_n$ on $D_n$. Prove that the number of conjugacy classes of the reflections are $\begin{cases} 1 &\text{ if } n=\text{odd} \\ 2 &\text{ if } ...
8
votes
1answer
121 views

on finite abelian groups

Let $G$ be a finite abelian group and let $M(G)$ be the set of all elements of $G$ that fix with any automorphism of $G$. Then prove $$M(G)=\langle1\rangle \text{ or } Z_{2}$$ Attempt: We know that ...
8
votes
1answer
84 views

Prove that $Tx=x^{-1}~\forall~x\in G.$ [duplicate]

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$
2
votes
2answers
247 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
votes
4answers
67 views

Let $\langle a \rangle $ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism.

I'm stuck on this proof. I need to prove: Let $\langle a \rangle $ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism. And this is the ...
2
votes
1answer
57 views

About finite $p$-group finitely generated.

Let $p$ a prime number. Let $G$ be a $p$-group finitely generated, say, $G = \left<x_1, x_2, \ldots, x_m \right>$. Denote by $\gamma_i(G)$ the $ith$ term of the lower central series of $G$. ...
1
vote
2answers
66 views

Question about Sylow's theorems and a particular group of order 60

I have a finite group $G$ with the following data: Its order |$G$| is 60, it has exactly 6 Sylow-5-subgroups $P_i$$\ $ (i=1,...,6) and |$N_G(P_i)$|=10 $\forall$ i. I have the following questions: ...
2
votes
3answers
79 views

Show that A is a field which has $9$ elements

For $A=\left\{\left( {\begin{array}{*{20}{c}} a&b\\ { - b}&a \end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
5
votes
1answer
95 views

Induction from normal subgroup, problem with degrees

Suppose $K$ is an arbitrary field, $G$ a finite group and $N$ a normal subgroup. If one know all the irreducible representations of $N$ and then form the induced representations, one can use Mackey ...
4
votes
1answer
162 views

Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
3
votes
1answer
149 views

Orders of centralizers $C_G(g)$ in a group of order 60?

Given a group $G$ of order 60 with 24 elements of order 5, 20 of order 3, and 15 of order 2, how do we find the sizes of centralisers of elements of $G$ without proving $G\simeq A_5$? By considering ...
1
vote
3answers
102 views

$\mathrm{Aut}(\mathbb Z_{11})$ is isomorphic to $\mathbb Z_{10}$.

The question is this: Prove that $\mathrm{Aut}(\mathbb Z_{11})$ is isomorphic to $\mathbb Z_{10}$. I tried to construct a mapping from $f\colon\mathbb Z_n\to \mathbb Z_n$ and $f([k])=[ka]$ where ...
1
vote
1answer
183 views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by ...
5
votes
1answer
509 views

Why is $S_5$ generated by any combination of a transposition and a 5-cycle?

Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
0
votes
1answer
93 views

Right cosets of $K=\{1,17\}$ in $U_{32}$

How can we list the distinct right cosets of $K=\{1,17\}$ in $U_{32}$, the set of positive integers relatively prime to $32$?
2
votes
3answers
116 views

Show that for G be a abelian group, $g \in G, g^m=1$, where $m=|G|$

I am pretty sure there is a name for the following theorem, but unfortunately, I don't known it. Theorem. Let $G$ be a abelian group with $m=|G|$, than for any $g \in G, g^m=1$. Currently I don't ...
1
vote
2answers
179 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
4
votes
2answers
58 views

Existence of element of order $l$ dividing the order of the group

In this post: Order of kernel of a homomorphism , someone say that since $l$ divides $m$, we can say that there exists some element $x \in \ker (\varphi)$ such that $o(x)=l$. But why is it true? I ...
0
votes
1answer
62 views

How I can prove tht $T$ is isomorphic to a finite set of natural numbers?

Let $T$ be a finite abelian group. We can consider $T$ the as group $ℤ/nℤ$ or $ℤ/qℤ×ℤ/mℤ$. My question is: How I can prove tht $T$ is in bijection with a finite set of natural numbers? That is, I ...
0
votes
2answers
58 views

“Lifting the centralizer”

Let $G$ be a finite group, $T\le G$ and $N\unlhd G$ with $(|N|,|T|)=1$. Clearly $T$ acts by conjugation on $G$ and $N$ is a $T$-invariant subgroup; for this reason $T$ induces naturally an action on ...
5
votes
2answers
192 views

Upper bound for the sum of the orders in a finite group

This is a cute question that I found in ML. I thought a little bit about it and couldn't complete it to a solution. Let $G$ be a non-cyclic finite group of order $n.$ Let $S(G)=\sum_{g \in G} ...
3
votes
2answers
978 views

A group of order 30 has a normal 5-Sylow subgroup.

There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me. Lemma Let G be a group of order 30. Then the 5-Sylow subgroup of G is normal. ...
1
vote
2answers
462 views

Orders of elements in cyclic groups

I think I'm a bit confused about the order of elements in cyclic groups. If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, ...
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votes
2answers
70 views

Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$

I am trying to understand the proof of the following proposition: A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series. ...
4
votes
2answers
75 views

Frattini Subgroup of p-Groups

Letting $P$ be a $p$-group and $\Phi(P)$ be the Frattini subgroup of $P$ (the intersection of all maximal subgroups), the challenge is "Prove that $P/N$ is elementary abelian implies $\Phi(P)≤N$" ...
2
votes
3answers
50 views

Computing $\langle (13746) \rangle$ in $S_7$.

How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
5
votes
5answers
105 views

How to prove $x \in H$

How to prove that Let H be a normal subgroup of a finite group G. If $\gcd(|x|, |G/H|)$ = 1, show that $x \in H$.
4
votes
3answers
327 views

Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$

Let G be the multiplicative group $\mathbb{Z}_n^*$ for $n = 2^k$ and $k \ge 3$. Can we prove that no element has order bigger than $2^{k-2}$ ? My solution (not really a solution) : Since $n=2^k$, I ...
3
votes
1answer
87 views

Conjugation on subgroups of $A_4$ faithful?

Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$ I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
2
votes
0answers
50 views

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
0
votes
1answer
50 views

Finding a subgroup of Multiplicative group $\mathbb Z_{32}$

I am backing on some basic points about the multiplicative groups, like $\mathbb Z_{32}$ ,to review and I am really in a bad confusion to write the elements of a subgroup of it. For example, I want to ...
-1
votes
1answer
45 views

What is the order of an element in $ℤ/2mℤ×ℤ/2ℤ$?

I know that the order of every $T∈ℤ/nℤ$ divides the size of the group $n$. My question is: What is the order of an element in $ℤ/2mℤ×ℤ/2ℤ$?
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vote
2answers
158 views

Group theory problem to be solved?

Let $G = S_n$, the symmetric group of order $n$, acting as permutations on the set $\{1,2,\dots,n\}$. Let $H = \{\sigma \in G \mid n \cdot \sigma = n\}$. (i) Prove that $H$ is isomorphic to ...
5
votes
2answers
146 views

Prove that $S_4$ has no subgroup isomorphic to $Q_8$.

The question is to prove that $S_4$ has no subgroup isomorphic to $Q_8$. Here is an answer. But what "then $H$ also contains all products of two 2-cycles" means in that answer? Thanks.
3
votes
1answer
79 views

Groups and Lagrange theory

There are two subgroups $H_1$, $H_2$ of $G$, if $H_1\neq H_2$ then $\gcd(|H_1|,|H_2|)=1$. Prove that the order of $G$ is a prime number and the group is cyclic. I know from Lagrange that the order ...
-1
votes
2answers
103 views

Cyclic Group of order $8$

Let $G=(a)$ be a cyclic group of order $8$ and let $H=(a^4)$ be its subgroup of order $2.$ Find the coset representation of $G$ by $H$.
0
votes
1answer
63 views

Prove if $g$ is an element of order $d$ and $d$ divides $n$ then $gn = 1$. [duplicate]

Prove if $g$ is an element of order $d$ and $d$ divides $n$, then $gn = 1$.
5
votes
3answers
751 views

Show that $N$ is a normal subgroup in $G$ when $N$ is the intersection of normal subgroups in $G$

QUESTION : Let $G$ be a group, let $X$ be a set, and let $H$ be a subgroup of $G$. Let $$N = \bigcap_{g\in G} gHg^{-1}$$ Show that $N$ is a normal subgroup of $G$ conitained in $H$. MY ATTEMPT: I ...
1
vote
2answers
869 views

Concatenation of 2 finite Automata

I have some problems understanding the algorithm of concatenation of two NFAs. For example: How to concatenate A1 and A2? A1: ...
1
vote
0answers
125 views

Order of kernel of a homomorphism

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
5
votes
3answers
264 views

Does $G\times K\cong H\times K$ imply $G\cong H$?

Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$?
3
votes
1answer
46 views

Maximal Subgroups Containing given Element

Let $G$ be an elementary abelian $p$-group of finite rank, and $1\neq g\in G$. How do we parametrize the maximal subgroups of $G$, which contain $g$?
1
vote
3answers
174 views

Direct product of finite cyclic groups of coprime orders [duplicate]

The Question is this: How many generators are there of the group $G\times H$, if $G$ and $H$ are cyclic groups of order $m$ and $n$, which are coprime? Let's say that $G$ is generated by $g$, and ...
2
votes
2answers
59 views

Verifying homomorphism $S_3 \to \langle \phi \rangle$

Let $$ \phi : \begin{array}{c} 1 \mapsto 2 \\ 2 \mapsto 1 \\ 3 \mapsto 3 \\ \end{array} \qquad \text{and} \qquad \psi : \begin{array}{c} 1 \mapsto 2 \\ 2 \mapsto 3 \\ 3 \mapsto 1 \\ \end{array}. $$ ...
0
votes
1answer
154 views

Prove that [GxH : AxB]=[G:A][H:B] when A < G and B < H

The original question is that: If A is subgroup of group G and B is a subgroup of group H, then express [GxH : AxB] in terms of [G:A] and [H:B] and prove the result is correct! Then I first prove ...
2
votes
2answers
166 views

Sylow Theorems Question

Let $p$ be the smallest prime that divides the order of a finite group $G$. If $H$ is a Sylow $p$-subgroup of $G$ and is cyclic. Prove that $N(H)=C(H)$.
-2
votes
2answers
104 views

Elements of a given order in finite cyclic groups

List all elements of order $4$ in $\mathbb{Z}_8=\mathbb{Z}/8\mathbb{Z}$. Also list all the elements of order $6$ in $\mathbb{Z}_{72}=\mathbb{Z}/72\mathbb{Z}$.