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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Groups of orders 108, 120, 144, 168, 180, 228 and 240

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 $240$.
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$N$ 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 ...
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1answer
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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?
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1answer
54 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?
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1answer
38 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 $\gcd(|G|,n)=1$?
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1answer
25 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 ...
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2answers
72 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 ...
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33 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 $G$ ...
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2answers
31 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 ...
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1answer
28 views

Complement but not direct summand

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.) ...
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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 ...
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1answer
42 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 ...
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2answers
42 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, ...
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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 ...
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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. ...
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1answer
39 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 ...
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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 ...
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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 $G=X_1X_2$?
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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 ...
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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.
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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)$.
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1answer
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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 ...
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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 ...
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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 ...
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1answer
41 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 $PN=G$
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1answer
49 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 $G$?
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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 ...
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0answers
30 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 ...
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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 ...
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1answer
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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.
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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 ...
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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 ...
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2answers
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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 $\sigma=(1\ 3\ 5\ 7\ 9)$?
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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 ...
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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} ...
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2answers
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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
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1answer
53 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 ...
2
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1answer
90 views

Sylow theorem Cyclic sylow p subgroup

$p$ is the smallest prime dividing order of $G$. $P$ is a sylow p subgroup which is cyclic. Prove that $N_G(P) = C_G(P)$ This is my approach : Since $P$ is sylow p subgroup so its order is some power ...
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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 ...
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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 ...
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56 views

Existence of a finite group union of self normalizing subgroups

Does a finite group G union of self normalizing subgroups such that the intersection of any two of these subgroups is equal to the unit of group G exist? I don't think so, but I can't prove it. Thank ...
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1answer
68 views

About $ AGL(1,16)$

I wanted to know orders of all subgroups of the group $AGL(1,16)$ (of order 240). Indeed, regarding to the problem mentioned in ...
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1answer
41 views

$C_{S_4}(A_4)=1$

I have to show that $C_{S_4}(A_4)$ is trivial. Now we know that $$ C_{S_4}(A_4)=\{x\in S_4\;:\:yx=yx\;\;\forall y\in A_4\}=\bigcap_{y\in A_4}\{x\in S_4\;:\;yx=xy\}\;. $$ Then every element $\neq1$ ...
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1answer
41 views

Do normal group endomorphisms form a normal submonoid?

What it says on the tin. A group endomorphism $v\colon G\to G$ is called normal if $v(aba^{-1})=av(b)a^{-1}$ for all $a,b\in G$. Equivalently, the map $g\mapsto v(g^{-1})g$ is a group homomorphism. ...
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2answers
66 views

$D_6$ is not a subset of $D_8$

I came across an example in Chapter-2 of Dummit and Foote(page-47) which says :$D_6$ is not a subgroup of $D_8$ ,the former is not even a subset of latter.I can't understand why is it not the subset ...
3
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1answer
81 views

Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
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1answer
38 views

“Powers” of injective representations “contain” all irreducibles

Let $G$ be a finite group and let $\rho : G \to GL(V)$ be an injective representation. I need to prove that each irreducible representation of $G$ is contained in $\otimes_{i=1}^{n} \rho$ for some $n ...
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1answer
64 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
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1answer
35 views

Subgroups of Symmetric groups isomorphic to dihedral group

Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n>2$?
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39 views

Generating series - Finite groups of order $n$

I am wondering if something of interest can be said about one of the two series $$G_1(x)=\sum_{n=1}^{+\infty}{\mathcal{G}(n)z^n}$$ $$G_2(s)=\sum_{n=1}^{+\infty}{\frac{\mathcal{G}(n)}{n^s}}$$ where ...