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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Show that the following group action has a non-zero singleton orbit.

Let F be a finite field of characteristics prime p.Let G be a group of order $p^r$ for some r.Let G acting on $F^n$ for n>1.Then show that there exist a non-zero vector in $F^n$ whose orbit will be ...
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1answer
29 views

Isomorphism with Euler phi function

Let $m_i > 1$, where $1 ≤ i ≤ n$, be integers, pairwise relatively prime. Let $m = m_1 \cdots m_n$. Let $\phi(m)$ denote the order of the group $(Z/mZ)^×$. The function $\phi : Z_+ → Z_+$ is called ...
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0answers
29 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
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1answer
35 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
28 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
21 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
36 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 ...
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2answers
28 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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1answer
21 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 ...
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1answer
14 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 ...
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2answers
66 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 ${\mathbb{Z}_{3}} ^{3}$ ...
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1answer
48 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
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1answer
56 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 ...
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2answers
89 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 $|G| \vdots 2$. ...
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0answers
29 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 ...
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3answers
171 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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3answers
37 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
12 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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0answers
9 views

Simplest way to see that the affine isometries of a regulara $n$-gon are linear?

What is the simplest way to see that the set of affine isometries of the plane that fix a regular $n$-gon centered at the origin are in fact linear? One can see this by showing that the origin is the ...
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1answer
45 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 ...
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40 views

Finding the group structure of a finite ring

Trying to construct an example I built up this finite ring: $$B=\mathbb{Z}/9\mathbb{Z}[x,y,z,w_1,w_2]/(x^3-1,y^3-1,(x-1)(z+3w_1),(y-1)(z+3w_2),w_1^2,w_2^2,z^2)$$ I need to know the structure of ...
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1answer
62 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 $D_8$?
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1answer
34 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 $h_1 \in H_1$, ...
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0answers
38 views

Expand group from it's presentation

I want to know if there is a method to expand a group given it's presentation, i.e. list all elements of the group. For instance $G = < x, y \ | \ x^2y = xy^3 = 1>$ (You don't need to solve ...
2
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1answer
31 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 = ...
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1answer
34 views

What is the center of the Valentiner group $\mathcal{V}=\langle I, Q \rangle$?

(Please refer to this question first: Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$? ) I want to understand the center of the Valentiner group: $$\mathcal{V}=\langle I, Q ...
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1answer
36 views

Finding a subgroup of $Z_4⊕Z_2$ that is not of the form $H⊕K$ [closed]

Find a subgroup of $Z_4⊕Z_2$ that is not of the form $H⊕K$, for $H$ a subgroup of $Z_4$ and $K$ a subgroup of $Z_2$ . Please help
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1answer
34 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) ...
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1answer
51 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 )
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3answers
46 views

Is a nonzero number infinitely greater than zero? [closed]

So many years ago, I posted this question on Yahoo! Answers, and was not really happy with the response. I ran across it again recently and decided to try and breathe new life into this in the hopes ...
3
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1answer
70 views

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

My question is the one in the title. If you want to understand the context of the problem, please read further. I reduced a problem to proving the question. Background is: Valentiner group ...
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0answers
22 views

suppose $G$ is a finite group.if every proper subgroup of $G$ is nilpotent then $G$ is solvable.

suppose $G$ is a finite group.if every proper subgroup of $G$ is nilpotent then $G$ is solvable. my answer:suppose $G$ is smallest group which is satisfied the condition of the problem below but not ...
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0answers
22 views

$G$ is a finite solvable group ,then there exist prime $p$ if $H$ is a Hall subgroup of $G$ that $p \nmid|H|$ then we have $H \lneqq N_{G}(H)$.

suppose $G$ is a finite nontrivial solvable group ,then there exist prime $p$ which if $H$ is a Hall subgroup of $G$ that $p \nmid|H|$ then we have $H \lneqq N_{G}(H)$. my work:we know that $G$ has a ...
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1answer
36 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 ...
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2answers
49 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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0answers
34 views

Possible number of Groups order N

For when is the number of groups of some order n more than n? For example, let say this happens at $n=3$ then that would mean that there are more groups of order 3 than 3.
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1answer
36 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 ...
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0answers
32 views

Looking for example of particular type of proper subgroups of $\big(\mathcal P(A), \Delta \big)$ [closed]

i) Let $A$ be a set with more than two elements , give example of a proper subgroup $H$ of $\big(\mathcal P(A), \Delta \big)$ such that $H$ has more than two elements ii) Let $A$ be an infinite set ...
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0answers
58 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
32 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
47 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
59 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 $H$ ...
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2answers
48 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 $p=3$ ?
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0answers
26 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
22 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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0answers
21 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
67 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 ...
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0answers
29 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
62 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 ...
0
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1answer
53 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 ...