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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3
votes
0answers
27 views

Product of the elements cannot be 1

I have to show in a group $G$ of order $4k+2$ the product of the elements cannot be $1$. What I got so far is that there exists a subgroup $H$ of index $2$ in $G$ (considering left regular ...
1
vote
0answers
13 views

(Proof-Strategy) Showing that the order of an element in the symmetric group is the least common multiple of the cycle lengths.

This question is not looking for a proof. I just want to make sure my proof-strategy makes sense and could be used to prove the following statement: Prove that the order of an element in $S_n$ ...
2
votes
2answers
31 views

Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
2
votes
0answers
26 views

Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
3
votes
1answer
41 views

There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
0
votes
2answers
33 views

Give an example of three different groups with eight elements

Give an example of three different groups with eight elements. Why are the groups different? One particular answer that I found was the groups $\mathbb{Z}_8$, $\mathbb{Z}_4 \times\mathbb{Z}_2$, and ...
9
votes
2answers
286 views

Is a group uniquely determined by the sets {ab,ba} for each pair of elements a and b?

For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function $$ F_G:S\times S\to\mathcal P(S)\\ F_G(a,b):=\{a\cdot b,~b\cdot a\} $$ from $S\times S$ ...
1
vote
1answer
43 views

Sylow subgroups problem

Let $G$ be a finite group and $p<q$ such that $p^2$ doesn't divide $|G|$. Let $H_p$ and $H_q$ be Sylow subgroups of $G$ with $H_p \lhd G$. Show $H_pH_q \lhd G \space \implies H_q \lhd G$. From the ...
1
vote
1answer
35 views

Sylow p-subgroups and Sylow theorem

Find all Sylow 3-subgroups of $S_3\times S_3$? This is what I already found: Since $O(S_3\times S_3)=36=2^2 3^2$ Sylow- $3$ subgroups have order $9$. If $n_3$ is the no. of Sylow- $3$ subgroups, Then ...
0
votes
1answer
24 views

Is this set a generating set for this (normal) subgroup?

Suppose $G$ is a finite group and $N$ is a normal subgroup of $G$. Suppose also that $X$ is a generating set for $G$, and that $Y$ generates $N$ as a normal subgroup of $G$ (i.e. $N$=$\langle ...
3
votes
2answers
83 views

Presentations representing different groups.

Using GAP, I knew that groups $G=\langle x,y;x^4,x^2y^2,xyxy^{-1}\rangle$ and $H=\langle x,y;x^4,y^4,xyxy^{-1}\rangle$ are different. But I want to prove it. I tried to do something using Tietze ...
1
vote
0answers
45 views

Groups of orders 108, 120, 144, 168, 180, 228 and 240 [on hold]

Prove that every group of order $\in \{ 108, 120, 144, 168, 180, 228\}$ has a subgroup of order or index $6$. But, this is not true for all groups of order $240$.
0
votes
1answer
24 views

Why if H⊴G and H is solvable and G/H is solvable, then G is also solvable?

Plus, to the proof any p-group G is solvable,although Z(G) is normal to G and which is abelian, but G/Z(G) is not abelian, so what is the chain subgroup to show that G is solvable?
1
vote
2answers
44 views

$N$ a normal subgroup of $G$ with $|G|$ odd and $|N|=5$ satisfies $N \subset Z(G)$

Exercise Let $N$ be a normal subgroup of $G$. Suppose that $|N|=5$ and that $|G|$ is odd. Show that $N \subset Z(G)$. I am sorry for not writing any work of mine but I really don't know where to ...
1
vote
1answer
58 views

I want to form a group of order one, is it possible?

Can a set with one element be a group, or does a group need to have at least two elements?
2
votes
1answer
26 views

Calculating number of conjugacy classes for a prime power group $G$

$\space$Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$. Since $Z(G) \neq G$, then $G$ is not abelian. I'll ...
1
vote
1answer
40 views

$x \rightarrow x^n$ is a group automorphism of a finite abelian group G [on hold]

How do we prove that the map $\phi:G \rightarrow G$ defined by $\phi(x) = x^n$ for some $n \geq 0$ is a group automorphism of $G$ if $\gcd(|G|,n)=1$?
1
vote
0answers
39 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
2
votes
2answers
37 views

If $|G|<\infty$ and $H\leq G$ is such that $[G: H]=2$ then $|x^G|=|x^H|$ or $|x^H|=\frac{1}{2}|x^G|$ for all $x\in H$?

Let $G$ be a finite group and $H$ a subgroup of $G$ with index $2$, that is, $[G: H]=2$. Recall that $$C_H(x)=H\cap C_G(x), $$ where $C_G(x)=\{g\in G: gx=xg\}$. How can I use the second isomorphism ...
1
vote
0answers
28 views

$G$ a finite group $n$-abelian goup and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then to show $G$ is abelian [duplicate]

Let $G$ be a finite group and $n$ be a given positive integer such that $(ab)^n=a^nb^n , \forall a,b \in G$ and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then how to prove that $G$ is abelian ? If I can show ...
1
vote
1answer
60 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
2
votes
1answer
45 views

Transitive action on two sets! [duplicate]

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of ...
1
vote
0answers
39 views

Intuition fail with normal subgroups

My intuition fails me, when I try to undestand normal subgroups. I read that Alternating group $A_n$ is simple for all $n \geq 5$, $n$ is the order of the group. So $A_4$ is possibly (this has to do ...
1
vote
2answers
43 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
0
votes
0answers
23 views

Crossed homomorphism: $\varphi(x)=\varphi(y)\iff Kx=Ky$

Let $G$ be a finite group, $N\unlhd G$. A crossed homomorphism from $G$ to $N$ is defined as a $\varphi:G\to N$ s.t. $\varphi(xy)=\varphi(x)^y\varphi(y)$. It's not in general a group homomorphism. ...
3
votes
2answers
81 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
3
votes
1answer
54 views

An equality from Representation Theory

Studying Representation Theory of finite groups I've bumped in the following identity: $$\frac{n(n+1)}{2}=\sum_{i=1}^n\frac{(2i-1)!!(2n-2i+1)!!}{(2i-2)!!(2n-2i)!!}$$ My book suggests to prove it ...
2
votes
1answer
40 views

Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
0
votes
1answer
50 views

Normal subgroup of a finite group

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?
1
vote
1answer
37 views

Rapid question on minimal normal subgroups

Why does $N$ finite group, $M\unlhd N$ imply that $M$ contains a minimal normal subgroup of $N$? If $M$ is itself minimal in $N$ we have finished. But if not, we know that there exists a minimal ...
0
votes
0answers
16 views

Index of every maximal subgroup of a solvable group is a prime power

G is a finite solvable group.Then the index of every maximal subgroup is prime power.Plz help me.
1
vote
1answer
34 views

question on nilpotent group.

Question- If $G$ is a finite group, $N$ a normal nilpotent subgroup of $G$ such that $G/[N,N]$ is nilpotent. Prove that $G$ is nilpotent. How i did it in my exam today- (I know my solution had to be ...
3
votes
1answer
55 views

Hecke Operators

I'm currently working my way through the section on Hecke operators in Serre's book. In the proof that $ T(m)T(n) = T(mn) $ for all $ m,n \in \mathbb{Z}_{\geq 1} $ with $ \textit{gcd}(m,n) = 1 $. In ...
3
votes
1answer
50 views

Direct product of two groups.

Let $N$ be a minimal normal subgroup of $G$. Also let $N$ and $\frac{G}{N}$ are non-abelian simple. Can we say that $G=N\times A$ where $A$ is a non-abelian simple subgroup of $G$?
1
vote
1answer
42 views

non-abelian simple group

Let G be a non-solvable group and $(\frac{G}{Z(G)})^{'}=\frac{G}{Z(G)}$. Can we say that there is a normal subgroup $N$ of $G$ with property $Z(G)\leq N$ such that $\frac{G}{N}$ is a non-abelian ...
1
vote
2answers
40 views

Let $ G$ be a nilpotent group. prove that there exist $a\in G$, such that $ o(a)=exp(G)$.

Please hint me. I want to proof the following homework: Let $ G$ be a nilpotent group. prove that there exist $a\in G$, such that $ o(a)=exp(G)$.
1
vote
1answer
43 views

If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$

If $G$ is a finite group and its $p$-sylow subgroup $P$ lies in the center of $G$, prove that there exists a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$
2
votes
3answers
83 views

Is there a simple group of order $105$?

By Sylow theorem, we see the number of the $7$-sylow subgroup is $n_7$. Then $n_7=1$ (mod $7$) and $n_7$ divides $15$; thus $n_7=15$, but why do we have $6\cdot 15=90$ elements of order $7$? And just ...
1
vote
0answers
17 views

How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up ...
-4
votes
1answer
49 views

mapping $S_4$ a nd checking if it is cyclic [duplicate]

True or false? Every element of $S_4$ is a cycle. Can anyone help me in how to solve this question? I find difficulty in answering the question.
2
votes
1answer
38 views

prove that the no of normal subgroups of a fixed order containing K is congruent to 1 mod p.

Let G be a p group K is a normal subgroup of G of order p to the power a. Then prove that the no of normal subgroups of a fixed order containing K is congruent to 1 mod p. I've proved that the no of ...
-1
votes
0answers
31 views

classify all groups $G\rtimes (\mathbb Z/p\mathbb Z)$

Let $G$ be a group of order $pm$ where $p$ does not divide $m$. i) If $G$ has a unique Sylow $p$-subgroup then classify all groups $G\rtimes (\mathbb Z/p\mathbb Z)$. ii) If $G$ does not have a ...
1
vote
1answer
86 views

When an Abelian group is cyclic

Let G be a finite abelian group.It contains a non trivial subgroup which is contained in every non trivial subgroup.Then G must be cyclic. This is a problem of Herstein book(Pg 108,#11 2nd edition).I ...
3
votes
2answers
49 views

Calculating number of elements commmuting with $\sigma\in S_{10}$

let $S_{10}$ denote the group of permutations on ten symbols ${1,2,3,....,10}$.Then how do we calculate number of elements of $S_{10}$ commuting with the element $\sigma=(1\ 3\ 5\ 7\ 9)$?
1
vote
0answers
33 views

Two Open Ended Questions in Sylow Theory

Sylow Theorems are very powerful in finite group theory. Two natural questions come to mind: 1) Given a finite group $G$ and a $p$-subgroup $H$ of $G$, how many Sylow $p$-subgroups of $G$ contain ...
1
vote
0answers
54 views

Automorphism Tower

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

Determining elements of group from generating relations

Is there an algorithm for determining all the elements fo a finite group from its generating relations? For example, let group $G$ have the generating relations $p^3 = q^2 = (qp)^2 = 1$. I see that ...
3
votes
1answer
39 views

A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.

I want to prove the following: Let $G$ be a group of order $2^nm$, where $m$ is odd, having a cyclic Sylow $2$-subgroup. Then $G$ has a normal subgroup of order $m$. ATTEMPT: We ...
2
votes
0answers
41 views

Group's morphisms

I know that the constant map equal to $1$ and the signature are two group's morphisms from the group of permutations $(\mathcal S_n,\circ)$ to the group $(\Bbb C^*,\times)$. My question is: Can we ...
3
votes
1answer
54 views

Generalization of a property of proper conjugate subgroups of a finite nilpotent group

Let $G$ be a finite nilpotent group and $H, K$ be two proper non-maximal subgroups of $G$ such that $H\not\leq\Phi(G), K\not\leq\Phi(G)$ and $\langle H, K\rangle < G$. Does there exist a maximal ...