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

2
votes
2answers
74 views

find the number of elements of order 3 in an abelian group of order 120

Let order of G=120. Then the number of sylow 3 grs are (1+3k)=p.where p divides 8. So k=0 or 1. Which one i take?
2
votes
1answer
62 views

Find the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n

What is the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n? This is a question in Joseph A.Gallian's book, and the answer is n=36 and the ...
1
vote
0answers
19 views

Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
4
votes
1answer
57 views

Can we always construct a “$p$th root” of a $p$-element in a finite group?

Let $p$ be a prime, $G$ a finite group, and $g\in G$ a $p$-element. Can one always embed $G$ in a finite group $H$ that contains a $p$-element $h$ such that $h^p=g$?
1
vote
1answer
28 views

Obtaining a presentation of the dihedral group from a semidirect product

I am working on classifying groups of order 44. I have shown that $G\cong P_{11} \rtimes_{\varphi} P_{2}$, where $P_p$ are Sylow p-subgroups and $\varphi:P_{2} \to $Aut$(P_{11})$ is a group ...
0
votes
1answer
45 views

if $G$ is finite group then polycyclic group is equivalent to super solvable group?

I don't know why this is true? can you help me: if $G$ is finite group then polycyclic group is equivalent to super solvable group Definitions- Polycyclic group $G$ is a polycyclic if has a ...
1
vote
1answer
51 views

Colouring a tetrahedron

How would I write down the elements sr and $sr^2$ of G as a product of disjoint cycles? If I am looking for the orbits of this action, do I have 4 orbits $\{1\:2\:3\}, \{p12\:p23 ...
3
votes
2answers
53 views

$G$ a finite group, $H$ a subgroup of index $2$ in $G$. If $K$ a subgroup of $G$ of odd order then $K$ contained in H.

Let $G$ be a finite group and $H$ a subgroup of $G$ such that $|G:H|=2$. Suppose $K$ a subgroup of $G$ of odd order. Show $K$ is contained in $H$. I'm stuck. Need a hint.
5
votes
2answers
61 views

Classification of all finite elementary $p$-groups.

Let $G$ be a finite group. For a prime number $p$, let us call $G$ an elementary $p$-group iff $\exp G=p$. I know that all elementary $2$-groups are abelian, and I also know the construction of ...
4
votes
0answers
50 views

Is a finite group which is generated by two fully invariant abelian subgroups always abelian?

Let $G$ be a finite group satisfying there exist two fully invariant subgroups $H,K$ of $G$ such that $H$ and $K$ are abelian and generate the whole group $G$. Then can we conclude that $G$ is ...
1
vote
1answer
45 views

under following conditions $G$ has only one subgroup of order $p$.

Let $|G|=p^m$ for $m \ge 2$. If every subgroup of $G$ of order $p^2$ is cyclic, then $G$ has only one subgroup of order $p$.
0
votes
0answers
31 views

Generaliation of semidirect product

Let $G=\langle X\mid R\rangle$ and $H=\langle Y\mid S\rangle$, and let $\phi :H \to \textrm{Aut}(G)$. Then the semidirect product $G\rtimes_{\phi}H$ has the following presentation: $G\rtimes_\phi H = ...
1
vote
1answer
27 views

Finding subgroups of $D_{12}$

Let $G=D_{12}=\langle a,b \mid a^6=b^2=e, bab^{-1}=a^{-1} \rangle$. Find all subgroups of $G$. We can easily spot the cyclic normal subgroup $C=\langle a \rangle = \{e,a,a^2,\dots,a^5\}$. Now ...
0
votes
2answers
38 views

homomorphism between $K = \langle u,v \mid u^2 = 1, v^4 =1, uv=vu\rangle$ and $ \mathbb{Z}_4$

Let $K = \langle u,v \mid u^2 = 1, v^4 =1, uv=vu\rangle$ be a group. Is there a homomorphism between K and $ \mathbb{Z}_4$?
1
vote
4answers
52 views

Order and index of a normal subgroup $N$ are relatively prime

Let $N$ be a normal subgroup of a finite group $G$. Assume that the order of $N$ and the index of $N$ in $G$ are relatively prime. Prove that if $g\in G$ satisfies $o(g)\mid o(N)$, then $g\in N$. ...
7
votes
1answer
55 views

If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple

I'm looking for verification: My claim: If $G$ is a finite group and $H$ is a (proper)subgroup of index $k>1$, where $k! \leq |G|$, then $G$ is not simple. Proof: Consider the set of left cosets ...
2
votes
1answer
30 views

Disjoint normal subgroups - one contained in the centralizer of the other

Let N and M be normal subgroups of a group G and assume that N and M have only one element in common. Prove that N is contained in $C_G(M)$. First I concluded that |NM|=|N|*|M|. Now I'm trying to ...
1
vote
2answers
56 views

Proving a quotient group is not Abelian without calculating actual cosets

Given the normal subgroup of S4: N={(1),(12)(34),(13)(24),(14)(23)}, show that S4/N is not Abelian. What I did was to calculate two random cosets of N in S4,like in the picture I attached, and show ...
2
votes
1answer
44 views

$H$ and $K$ are subgroups of $G$.Recall that $HK=\{hk:H\in H,k\in K\}$ Show that $|HK|=|H||K|/|H\cap K|$

Let $G$ be a finite group, and let $H$ and $K$ be subgroups of $G$.Recall that $HK=\{hk:H\in H,k\in K\}$ Show that $|HK|=|H||K|/|H\cap K|$
2
votes
3answers
52 views

Finding the order of Z(G) in a non-Abelian group of order 8 [duplicate]

Let $G$ be a non-Abelian group of order $8$. Prove that $|Z(G)|$ is less or equal to $2$. First I must say this is a question about normal subgroups. I haven't yet studied homomorphisms or more ...
1
vote
3answers
45 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
votes
1answer
94 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 ...
0
votes
1answer
35 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
votes
3answers
40 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
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 ...
2
votes
1answer
72 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) &= ...
0
votes
1answer
28 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
53 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
votes
2answers
54 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 ...
8
votes
2answers
203 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
votes
1answer
41 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
votes
1answer
46 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
votes
1answer
39 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
1answer
73 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
votes
2answers
61 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 ...
4
votes
4answers
127 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
38 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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
39 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
votes
2answers
47 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
votes
1answer
52 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
votes
2answers
81 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 ...
2
votes
3answers
129 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
96 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. ...
34
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 $H$ ...
-2
votes
1answer
24 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
17 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
43 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
35 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
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 ...