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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128 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
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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 ...
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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
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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
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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 ...
3
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1answer
99 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) &= ...
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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
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3answers
98 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
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2answers
137 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 ...
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2answers
222 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
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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
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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
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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
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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
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2answers
70 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
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4answers
405 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
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1answer
51 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 ...
2
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1answer
49 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
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2answers
110 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
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1answer
75 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
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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 ...
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3answers
815 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
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0answers
150 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. ...
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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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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 $C_2$ ...
0
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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
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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
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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
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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
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1answer
105 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 ...
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1answer
66 views

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question: Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. Can you give me some hints ...
1
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1answer
17 views

Proving the following subgroup(verification of logic)

So I was reading the following theorem from dummit that is If $|H| = n <\infty$ then for each positive integer dividing n there is a unique subgroup of $H$ of order $a$. This subgroup is the ...
2
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1answer
82 views

Is there a group homomorphism $f:G\longrightarrow G$ for which $G/\operatorname{Im} f \not\cong\operatorname{Ker} f$? $G$ is finite

Can you find a counterexample to the claim that for all group homomorphisms $f:G \longrightarrow G$, $G/\operatorname{Im} f \cong \operatorname{Ker} f$. Let $G = \mathbb{Z}$; $f(n) = 2n$ is a ...
2
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0answers
69 views

Need help in understanding the question. Elemntary number theory

I have this question in my home assignment. I contains two parts and I don't quite understand what is the difference between them.The question is: Let $n > 2$ be an integer such that ...
1
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1answer
26 views

question about isomorphism involving Dihedral group.

suppose $D_{n}$ is dihedral group with order $2n$, do we have this Isomorphism below? $$D_{2k+1} \times \mathbb{Z}_2 \simeq D_{4k+2}$$ I think it is wrong, I couldn't find any mapping, but I ...
4
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1answer
103 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
2
votes
1answer
35 views

Direct decompositions and quotients of abelian groups

Let $G = \langle a \rangle_{27} \oplus \langle b \rangle_{81}$. Find a direct decomposition $G = \langle 10a + 60b \rangle \oplus ?$. Find the elementary divisors of $G/ \langle 3a + 18b \rangle$. ...
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1answer
97 views

Herstein Problem No.7 Page 102

let $G$ be a group of order $30$ .How many non-isomorphic groups of order $30 $ are there? Before doing this I have shown that every Sylow 3 and Sylow 5 subgroup is normal in G and G has a normal ...
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0answers
58 views

Showing the existence of $O^{\pi}(G)$

Assume G is a finite group. I am trying to show the existence of $O^{\pi}(G)$, the unique normal subgroup of G minimal such that $G/O^{\pi}(G)$ is a $\pi$-group, i.e. a group whose order is divisible ...
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1answer
43 views

Sum of Inverses of the elements in $\mathbb Z_p^*$

If $p $ is an odd prime and if $1+\frac{1}{2}+\cdots +\frac{1}{p-1}=\frac{a}{b}$ where $a,b $ are integers prove that $p|a$. If $p>3\implies p^2|a$ My Try: Can the problem be interpreted as a ...
0
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1answer
47 views

information about semi-dihedral groups.

my question is about the elements and the generalized format of caylay table of groups called semi-dihedral groups which have the presentation $$ \langle a,b\mid a^{4m}=b^2=1,ab=ba^{2m-1}\rangle $$ ...
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0answers
27 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...
6
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2answers
87 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...
2
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1answer
104 views

Is there a cyclic subgroup C of S8 such that the interval lattice [C,S8] is distributive?

I've checked by hand that for any $n \le 7$, there is a cyclic subgroup $C$ of $S_n$ such that the intermediate subgroups lattice $\mathcal{L}(C \subset S_n)$ is distributive. Question: Is it the ...
0
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1answer
49 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...
0
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2answers
49 views

A subgroup of $\textrm{GL}(3,q)$ of order $q^2(q-1)$

Let $q$ be a prime power. Consider the multiplicative group $\textrm{GL}(3,q)$ of the $3 \times 3$ matrixes with coefficients in $\mathbb{F}_q$ which are invertible. The matrixes $$ M_{a,b,c} = \left( ...
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1answer
54 views

$S_4$ is not supersolvable? Why am I wrong?

I read that $S_4$ is an example of a solvable group who is not a supersolvable group. In order to prove it is solvable, we see that: $\{e\}<\{(1),(12)(34)\}<K<A_4<S_4$ where $K$ is the ...
1
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1answer
106 views

Product of a character with an irreducible character a non-negative integer

Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would ...
15
votes
4answers
703 views

Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...