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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54 views

A Question on the Quotient Group and/or set of cosets

I'm just confused about a somewhat simple fact about quotient groups. If we have: $$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we ...
2
votes
1answer
69 views

Showing a Group $G$ is not Simple [duplicate]

Let $G$ be a finite group of order $pq$, where $p,q$ are distinct prime numbers. Show that $G$ is not simple. Here is my attempt: $|G|=pq$. If $G$ is not simple, then it has non-trivial subgroups, ...
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2answers
76 views

To finde the center of $D_4$

is there a nice/smart way to find the center of $D_4$? rather then going through every element?
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1answer
44 views

Transfer homomorphism for abelian group.

If $G$ is abelian and $H \leq G$ of index $n$ , then show that transfer map is just $g \to g^n$. If i follow the definition from issac, transfer map will be same as pretranfer map as G is abelian ...
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0answers
46 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a ...
3
votes
3answers
98 views

Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups

I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| > 1$. There has been a similar question on MSE here: How to show ...
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vote
1answer
29 views

Showing that $|Z(G)| = p$ and $G/Z(G) \simeq \mathbb{Z}_p \times \mathbb{Z}_p$ for finite p-groups with order $|G|=p^3$

I have a finite, non-abelian $p$-group $G$ with $|G|=p^3$. I want to show that $|Z(G)| = p$ and $G/Z(G) \simeq \mathbb{Z}_p \times \mathbb{Z}_p$, where $Z(G)$ is the center of $G$. From the ...
0
votes
1answer
23 views

images of subgroups of $G = \mathbb{Z}_4 \times \mathbb{Z}_4$ in $G/G[2]$

I am asked which subgroups of $G = \mathbb{Z}_4 \times \mathbb{Z}_4$ have the same image in $G/G[2]$, where $G[2] = \{ g \in G: \operatorname{ord}(g) \,|\, 2\}$). I have determined all subgroups and ...
1
vote
2answers
116 views

Cyclic finite groups and their direct product

Let $C$ be a cyclic group with three elements. prove that $C \times C \times C$ can not be generated by two elements. I was thinking that showing that $C ^{3}$ is isomorphic to ${\mathbb{Z}_{3}} ^{3}$ ...
2
votes
1answer
121 views

Suppose that half of the elements of G have order 2 and the other half form a subgroup H of order n. Prove that H is an abelian subgroup of G.

Let $n>1$ be a positive integer. Let $G$ be a group of order $2n$. Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian ...
0
votes
1answer
176 views

The number of solutions of $x^n = e$ in a finite group is a multiple of n, whenever n divides the group order.

Prove that in a finite group G the number of solutions of the equation $x^n = e$ is a multiple of n, whenever n divides the order of the group. I feel there is a very simple answer to this question, ...
2
votes
1answer
125 views

Prove either $G=ST$ or |$G|\geq|S|+|T|$

Let G be a finite group, and let S and T be (not necessarily distinct) nonempty subsets. prove that either $G=ST$ or |$G|\geq|S|+|T|$ That's my thougt, I am thinking suppose $G$ does not equal to ...
3
votes
2answers
107 views

If the commutator of a finite group has order $2$, then the order of the group is divisible by $8$

Prove that if $|G| < \infty$ and $|G'| = 2$ then $|G|$ is divisible by $8$. Thoughts. $A \simeq G / G'$ is abelian and $G' \simeq \mathbb{Z}_2$. Since $G' \subset G$ then at least $|G| \vdots 2$. ...
2
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0answers
55 views

If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
14
votes
3answers
220 views

$|G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + \frac{|G|}{\left|\langle ab\rangle\right|}$

Show that for every finite group $G$ and for every elements $a, b \in G$ the following expression $$ |G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + ...
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vote
3answers
68 views

What is a conjugacy class of reflection?

I have a problem to do, asking to show that $D_{2n}$ has two conjugacy classes of reflections if n is even, but only one if n is odd. My question is, what is a conjugacy class of reflection? I have ...
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0answers
107 views

Find generators in GF(19)

I have 2 questions. Finding generators in GF(19) is similar to finding generators in GF(2^p)? Is primitive polynomial needed to find generators for GF(19)? Thanks a lot. Ya Ali.
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1answer
73 views

$G$ is a primitive group

Let permutation group $G$ contains a minimal normal subgroup $\neq 1$ which is transitive and Abelian. Show that $G$ is primitive. My attempts: Because of Proposition 4.4. of Wielandt's book ...
3
votes
1answer
66 views

Is the following group either a quaternion group or $D_8$?

Let $|G|=2^n$ and $Z(G)=G'=\Phi(G)$ where $\Phi(G)$ is the Frattini subgroup and $|Z(G)|=2$. Is $G$ necassarily either a quaternion group or $D_8$?
0
votes
1answer
103 views

Equality of cosets implies equality of the original sets

Let $H_1$, $H_2$ be two subgroups of $G$ containing $K$, where $K$ is a normal subgroup of $G$. Then if $H_1/K = H_2/K$, prove that $H_1=H_2$. Attempt: Let $h_1K = h_2K$, for some $h_1 \in H_1$, ...
2
votes
1answer
124 views

How to partition a finite vector space into affine subspaces all of the same dimension

Given an $n$-dimension vector space $V$ over a finite field $\mathbb F_q$ and a natural number $d<n$, the goal is to write $V$ as disjoint union of $d$-dimensional affine subspaces $v_i+V_i$: $$V = ...
0
votes
1answer
78 views

Additive subgroups of the finite field GF($2^m$)

Consider the set $G=\left\{ {0,1,...,{2^m} - 1} \right\}$. The elements of this set can be viewed as the elements of GF($q=2^m$) with appropriate addition/multiplication operations. For example, GF(4) ...
5
votes
1answer
153 views

every group of order $105$ has a cyclic normal subgroup of index $3$ ?

Does every group of order $105$ has a cyclic normal subgroup of index $3$ ? (Please don't use Sylow theorems )
3
votes
1answer
86 views

Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?

My question is the one in the title. Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?
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1answer
91 views

Alternate Form of Correspondence Principle

Let G be a group and K a normal subgroup in G. Let f : G → G/K be the canonical epimorphism given by x 􏰁→ xK and L a subgroup of G/K. Then (1) There exists a subgroup H of G. H contains K, and ...
2
votes
2answers
198 views

Groups of order 24.

I supposed $n_3=4$ and $n_2=3$, and then I made $G$ act by conjugation on $Syl_3 (G)$. I want to show that $G\cong S_4$ (looking at all order 24 groups here ...
2
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1answer
88 views

Possible number of groups of order N

When is the number of groups of some order $n$ greater than $n$? For example, lets say this happens at $n=3$, then that would mean that there are more groups of order 3 than 3.
2
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1answer
41 views

A commutator relation

I hope the following is not trivial, Let $H$ be a subgroup of $G$ s.t. $[H,G]\leq Z(G)$ then can we say that $H$ is normal ? I think we can not but I could not find counter example. Any counter ...
9
votes
0answers
64 views

A hard question on surjective group homomorphism [duplicate]

Say $G$ and $H$ are finite groups, and there exists a surjective group homomorphism from $G × G$ to $H × H$. Must there exist a surjective group homomorphism from $G$ to $H$? I have no idea how to do ...
2
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1answer
41 views

A specific group lacking Følner sequences

How does one go about proving that the free group $<a,b,a^{-1},b^{-1}>$ lacks any Følner sequence?
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1answer
89 views

If G is finite p-group and H is a subgroup, show that there is a composition series that contains H.

Let $p$ be a prime number, suppose $G$ is a finite p-group and let $H$ be a subgroup of $G$. Show that there is a composition series that contains $H$. I have already shown that if $G$ if a finite ...
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0answers
67 views

Is this subset of group, a subgroup of it? [duplicate]

Suppose that $G$ is a finite group, $n = |G|$, and $X$ be a non-empty subset of $G$. Is it true that $$ H := \{ x_1 x_2 \dots x_n | x_i \in X \} $$ is a subgroub of $G$? Edit: Since $n=|G|$ so $H$ ...
2
votes
2answers
58 views

A question on finite group

Let $G$ be a finite group and $p$ be the smallest prime divisor of $|G|$ , let $x \in G$ be such that $o(x)=p$ , and suppose for some $h\in G $ , $hxh^{-1}=x^{10}$ , then is it true that $p=3$ ?
3
votes
0answers
48 views

On transitive constituents of a permuation group

Assume that the intransitive permutation group G has degree n and minimal degree n−1. If no transitive constituent of G has degree 1, then they all are faithful and all except one are regular. I ...
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0answers
31 views

Getting the least possible order of a group from existing cyclic subgroups or elements.

Let's say I have finite group $G$. Let it have have following subgroups: $<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$ I can for sure ...
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votes
0answers
25 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
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1answer
89 views

showing that $G$ is nilpotent.

suppose $G$ is a finite solvable group,then $G$ is nilpotent if and only if all Hall subgroups of $G$ which its indices are power of a prime number are normal. suppose $G$ is a solvable finite group ...
2
votes
1answer
58 views

showing that $G$ is not solvable.

suppose $G$ is a finite group and $1\neq a \in G$ ,$1\neq b \in G$ and $O(a)$ ,$O(b)$ ,$O(ab)$ every two of them are relatively prime ,then $G$ is not solvable. my Idea:I suppose $G$ is solvable and ...
3
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0answers
85 views

Sylow $p$-subgroup of $GL_{n}(\mathbb{F}_{q})$

This question has already been asked here. However, the answer there isn't particularly helpful. Consider $G = GL_{n}(\mathbb{F}_{q})$ where $q = p^{r}$ where $p$ is prime. Show that the upper ...
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votes
1answer
539 views

Question about Z12≃Z4⊕Z3 generators and order

I apologize in advance for my messy language and questions; I've only been studying group theory for a month and thus these concepts aren't clearly locked in yet; hence my questions. :) Take ...
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1answer
40 views

Character of the algebra $\mathbb{C}[G]$ as $G \times G $-module

Let $G$ be a finite group. We can define an action of $G\times G$ on the group algebra $\mathbb{C}[G]$ in the following way: If $x \in \mathbb{C}[G]$ then $(g,h)\cdot x=gxh^{-1}$. Now, what about ...
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2answers
187 views

Any group which is of prime order is a cyclic group

I don't know how to prove this: Any group which is of prime order is a cyclic group. What fact should I use to prove this?
2
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1answer
107 views

proof of a useful counting result in group theory

Let G be a finite group, H a subgroup of G satisfying |G| |̸| [G : H]!. Prove there exists a normal subgroup N of G satisfying 1 < N ⊂ H. maybe the General Cayley's Theorem works. I am not sure. ...
1
vote
1answer
99 views

Why is $G$ abelian?

If $|G|=pq^2$ with $p,q$ primes and if $p<q$, with $q\not\equiv\pm1\mod p$, why is $G$ abelian ? The $3^{rd}$ Sylow theorem implies that $n_p|q^2$ and $n_p\equiv 1 \mod p$, By hypothesis, ...
4
votes
3answers
59 views

if $k$ is a positive integer and $G$ a finite group such that $G=\{x^k:x\in G\}$ , then is it true that g.c.d.$(|G|,k)=1$ ?

If $G$ is a finite group of order $n$ and $k$ is a positive integer such that g.c.d.$(n,k)=1$ , then I know that $G=\{x^k:x\in G\}$ ; is the converse true ? that is if $k$ is a positive integer and ...
0
votes
2answers
54 views

$S$ nonempty finite subset of group $G$ for which $SS=S$. $S$ is subgroup.

Let $S\subset G$, $S$ finite and nonempty, $G$ group. Suppose additionally that $$SS=\{s_1 s_2: s_1\in S, s_2 \in S\}=S.$$ How can I prove that $S$ is a subgroup of $G$? Does this hold for $S$ ...
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2answers
97 views

Is $ G \cong G/N \times N$?

If G is a finite group and N is a normal subgroup in G , then can we say G $\cong$ G/N $\times$ N always? Is it true for like normal nilpotent or normal solvable or any such special classes. I ...
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votes
1answer
61 views

Automorphism Tower for $\Bbb Z_4$, $\Bbb Z_3$, $\Bbb Z_2 \times \Bbb Z_2$

Let $G$ be a group. Now consider $\operatorname{Aut} G$ then $\operatorname{Aut} (\operatorname{Aut} G) = \operatorname{Aut}(G^2)$, then $\operatorname{Aut}(\operatorname{Aut}(\operatorname{Aut} ...
3
votes
2answers
87 views

$G$ is product of its center and commutator

Let $G$ be a group s.t. $G=Z(G)G'$. When $G$ is abelian or perfect, the above equality is trivially true. We can also construct an example like $\mathbb{Z}_3 \times A_5$ i.e. a product of abelian and ...
9
votes
4answers
186 views

How many $A_5$ are there inside $A_6$?

I am reading a paper that says: There are $12$ versions of $A_5$ in $A_6$: $1) $ the permutations that leave one thing unmoved. $2)$ the permutations of the six pairs of antipodal ...