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.

learn more… | top users | synonyms

3
votes
3answers
29 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
13 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 ...
0
votes
1answer
19 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
35 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 ...
3
votes
1answer
30 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 ...
1
vote
0answers
27 views

For any subnormal subgroup of a finite group, must its normalizer be subnormal, too? [on hold]

Let $G$ be a finite group and $H$ a subnormal subgroup of $G$. Must $N_G(H)$ be subnormal, too?
8
votes
2answers
188 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 ...
2
votes
1answer
56 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 $S_4$ ...
0
votes
2answers
27 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 ...
1
vote
1answer
34 views

Groups occuring as derived subgroups.

I want to prove 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 ...
2
votes
2answers
43 views

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

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: (1)$G/N$ is a $p$-group (I guess it can be trivial group). ...
0
votes
1answer
34 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
3answers
58 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
34 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
votes
1answer
27 views

Center of group of a dihedral group

An example from my text ask to verify this: $$Z(D_{n})= \begin{cases} {R_{0},{R_{180}}} & \text{when n is even}\\ {R_{0}} & \text{when n is odd}\end{cases}.$$ How should I begin to verify ...
0
votes
2answers
69 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 ...
3
votes
2answers
39 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 ...
33
votes
3answers
1k 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 $H$ ...
1
vote
1answer
33 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 ...
1
vote
1answer
64 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 ...
2
votes
2answers
996 views

Possible Class equation for a group

Determine the possible class equation for a group of order 21? Until now I have found the following: $1+3+3+7+7$ $1+1+1+3+3+3+9$ $1+1+1+1+1+1+1+7+7$ $1+1+1+1+1+1+1+1+1+3+3+3+3$ $1+1+1+\cdots +1 ...
0
votes
1answer
46 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 ...
8
votes
2answers
183 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 ...
2
votes
3answers
103 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
83 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. ...
8
votes
1answer
103 views

Is $SL(2, 3) $ a subgroup of $SL(2, p)$ for $ p>3$?

As the title says, I was wondering whether $SL(2,3)$ is a subgroup of $SL(2,p)$ for $p>3$. I know that it is for $p=5$ (it can be found explicitly using the quaternionic representation), and I ...
-1
votes
1answer
18 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
votes
0answers
15 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
votes
0answers
38 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
23 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 ...
-3
votes
0answers
18 views

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. [closed]

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. is there theorem that said can let $ K = K_{\pi^{\prime}}K_{\pi^{\prime}}$ that $K_{\pi}$ is $\pi$-Hall subgroup of $ K $ and $ ...
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 ...
-1
votes
1answer
33 views

If $|G| = pqr$ for $p<q<r$ primes and all the Sylow groups are normal; is $G$ abelian? [closed]

Let $G$ be a group with $|G| = pqr$ for distinct primes $p<q<r$. If every Sylow subgroup of $G$ is normal, then is $G$ Abelian? Thank you in advance.
-1
votes
0answers
18 views

does dicyclic group of order $2^{\alpha}m$, $(2,m)=1$ have normal subgroup of order $m$?

consider dicyclic group with presentation $$Dic_n=< a,b |a^{2n}=1,a^n=b^2,ba=a^{-1}b>$$ the order of this group is $4n$,suppose that $|Dic_n|=2^{\alpha}m$ where $(2,m)=1$ my question is about ...
1
vote
1answer
53 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
3
votes
1answer
64 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 ...
1
vote
1answer
36 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 ...
-2
votes
1answer
58 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
vote
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 ...
1
vote
1answer
65 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
2
votes
0answers
63 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
vote
1answer
23 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
votes
1answer
68 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 ...
0
votes
1answer
77 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 ...
2
votes
1answer
27 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 $G/ \langle 3a + 18b \rangle$. ...
26
votes
5answers
3k views

Is there an easy way to see associativity or non-associativity from an operation's table?

Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot & a & b & c & d & e\\\hline a ...
4
votes
1answer
2k views

Cayley table help

I have a Cayley table with four elements and a binary structure $*$. I know that if I have the same element along the main diagonal (from top right corner to bottom left corner), then the set is ...
21
votes
5answers
4k views

Group of order 15 is abelian

How do I prove group of order 15 is abelian? Is there any general strategy to prove that a group of particular order(composite order) is abelian?
2
votes
0answers
57 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 ...
0
votes
1answer
31 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 $$ \langle a,b\mid a^{4m}=b^2=1,ab=ba^{2m-1}\rangle $$ ...