A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

learn more… | top users | synonyms (2)

3
votes
2answers
63 views

Want to clarify whether I am correct or not, $\Phi(G) \subseteq \Phi(H)$?

I Want you to clarify whether I am correct or not regarding following question. I will be thankful to you for telling me if I am wrong: Let $G$ be finite group and $\Phi(G)$ denotes its frattini ...
1
vote
1answer
186 views

Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Find all numbers $n$ such that $S_7$ contains an element of order $n.$ Identity is the only element of order $1.$So $n=1$ is possible. Case 1: Elements that can be written as a unique cycle of ...
7
votes
1answer
337 views

$H\leq Z(G)$ for a normal Subgroup $H\leq G$ Dummit Foote 4.4.12,13,14

Title is More General (which may not be true) But the Question is : Let $G$ be a Group of order $3825$. Prove that if $H\unlhd G$with $|H|=17$ then $H\leq Z(G)$. what i have done so far is ...
2
votes
1answer
57 views

Embedding and Graph(Tree) of Groups

According to Serre's definition (in Serre's Trees): ($G$,$T$) is a tree of groups if T is a tree and there are groups $G_v$ and $G_e = G_\bar e $ for each $v\in vertT$ and $e \in edge T$, where ...
3
votes
0answers
103 views

Converse of sylow theorem

I have been recently I asked to construct non isomorphic groups of specific order. To do that I used the invariant that they must have isomorphic centers. $G_1\cong G_2 \Rightarrow Z(G_1) \cong ...
0
votes
1answer
57 views

Problem on homomorphisms and fibres

I have a function $f: R^2$ to $R$ defined by $f(x,y) = y-x$. Let $H= [(x,x)|x ∈ R]$ and $H$ is a subset of $R^2$. Thinking of $R^2$ and $R$ as groups under vector addition: I need to check that $f$ ...
5
votes
4answers
2k views

Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.

Just would like to see if I'm right. Proof: Let $G$ be generated by $x$ (Let $G=\langle x \rangle$ for some $x$ in $G$.) We wish to show that there exists an $x'$ in $G'$ such that $G'=\langle x' ...
1
vote
1answer
76 views

Is this group isomorphic to $\mathbb{Z}_k$? The integers with multiples of $k$ subtracted until you're under $r$.

Where as the elements of the ring of integers modulo $n$ can be found by taking the integers and subtracting multiples of $n$ until you're just under $n$, consider the structure formed by taking ...
1
vote
3answers
284 views

Find order of $xy$ provided $x^2=e, y^3=e$ and $yxy=xy^2x$

Let $G$ be a group and $x,y \in G$. Suppose $x^2=e$, $y^3=e$ and $yxy=xy^2x$. Find the order of $xy$. I have no idea what to do. I am suspecting $(xy)^6=e$ (order 6) but I don't know how to solve ...
0
votes
2answers
90 views

Choosing “Conjugating element” from a subgroup

Let $H\leq G$ and $N\unlhd G$ such that $G=HN$ and $H\cap N= (1)$. My question is Prove that if two elements of $H$ are conjugate in $G$, then they are conjugate in $H$. What i have done so far ...
3
votes
1answer
146 views

If $aH = Hb$ for all $a,b \in G$, prove that $aH = Ha$.

If it holds for all $a, b \in G$ then substitute $b = a$ and the result follows. Right?! This was an old exam question; I've been staring at it for a while now and I simply can't believe it's ...
6
votes
1answer
399 views

sylow's theorem, Wielandt's proof.

Let $|G|=p^{n}m$, and let $0\leq k\leq n$. Then the number of subgroups of $G$ of order $p^k$ is congruent with 1 modulo $p$. I know Wielandt's proof in the case $k=n$, which is standard, but i dont ...
8
votes
0answers
238 views

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
0
votes
1answer
94 views

If size of each conjugacy class is atmost $2$ then $G'\leq Z(G)$

Question is : Show that if the size of each conjugacy class of a group $G$ is at most $2$ Then $G'\subseteq Z(G)$. Suppose size of conjugacy class of an element $g\in G$ is $1$ i.e., $ngn^{-1}=g$ ...
5
votes
1answer
560 views

Two elements that generate a group of order $22$

A group $G$ of order $22$ contains elements $x$ and $y$, where $x \neq 1$ and $y$ is not a power of $x$. Show that the subgroup generated by these two elements is the whole group. I'm not ...
5
votes
1answer
518 views

Proving the following conditions are equivalent

I have to prove four conditions are equivalent. I'm guessing I should proceed with (a) implies (b), (b) implies (c), (c) implies (d), and (d) implies (a)? I have gotten (a) implies (b), (b) implies ...
1
vote
0answers
198 views

Cayley theorem to prove that a group of order $2^mk$, $k$ odd, can't be simple

I have to solve this exercise WITHOUT Sylow theorems and Cauchy Lemma. In fact this exercise is given in the Cayley's theorem section. let $G$ be a group of order $2^mk$, where $k$ is odd. Prove ...
6
votes
1answer
199 views

What are the elements of order $n$ in symmetric group $S_n$?

Title explains it nicely :-) I'm interested in listing/generating them all for small n, up to n=12 or so.
0
votes
4answers
104 views

An innocent homomorphism between groups

There does not exist a nontrivial homomorphism $h:(\mathbb{Z}_8,+)\rightarrow(\mathbb{Z}_3,+)$. I am trying to understand this well-known fact. Let $h(x) = x$. Then certainly $h(a+b) = h(a) + ...
0
votes
1answer
407 views

What is a subdirect product?

I'm having trouble understanding what a subdirect product is. Say $G$ is a subdirect product of $H=\prod H_i$ - this means that the homomorphisms $f_i:G\to H_i$ are surjective, which can be ...
3
votes
3answers
123 views

Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
5
votes
5answers
115 views

Generator for $[G,G]$ given that $G = \left<S \right>$.

If $S'$ is a generating set for $G$, let $S=S' \cup \{s^{-1} \,| \, s\in S\}$, then is the set $[S,S] = \{[s,z] \,|\, s,z \in S\}$ a generating set for the commutator subgroup $[G,G]$? I want to ...
-1
votes
1answer
106 views

About a past question [closed]

In What is the center of a semidirect product, $\operatorname{Z}(G_1 \rtimes_\varphi G_2)$?, Alexander Gruber answered a question of user39794. Is there cited reference for this? I want to cite this ...
4
votes
2answers
286 views

Finitely presented Group with less relations than Generators.

I some how feel that, any finitely presented group with less relations than Generators has to be an infinite Group. In One of my Questions, In an answer I have seen a group which is finitely ...
2
votes
0answers
91 views

Proof of $G=N\rtimes H$ iff $G=NH$ and $N\cap H=\{1\}$

I have the following definition of a semi-direct product: Let $G$ be a group. Suppose $N\triangleleft G$ and $H<G$ such that every element of $G$ can be uniquely written $g=nh$. Then $G$ is the ...
2
votes
2answers
59 views

Group $G$ with $N\unlhd G$, $|N|=5$ and $G/N\cong S_3$

Group $G$ with $N\unlhd G$, $|N|=5$ and $G/N\cong S_3$. Question is : Let $G$ be a group with a normal subgroup $N\unlhd G$ of order $5$, such that $G/N\cong S_3$. Show that, $|G|=30$, $G$ has a ...
3
votes
1answer
95 views

How to determine all conjugacy classes in the complex orthogonal group $O(n,\mathbb{C})$ of finite order $k$

I was wondering how one would go about determining the conjugacy classes of the complex orthogonal group $O(n,\mathbb{C})$ of some finite order $k$. That is, if $[A]$ is the conjugacy class of $A\in ...
2
votes
2answers
49 views

Is the cartesian product of objects in an elementary topos cancellative?

My question is the internalization of this question to an elementary topos $C$. Is it true that: For objects $X,Y$ and $Z$ in an elementary topos $C$ with $X\times Y\cong X\times Z$, then also ...
1
vote
2answers
121 views

Normal automorphism of a perfect group

Show that the only normal automorphism (i.e., commutes with every inner automorphism) of a perfect group is the identity automorphism.
7
votes
0answers
268 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
2
votes
0answers
69 views

Is following true?

Is following true? Let $G=H \ltimes K$, where $H$ is a core free subgroup and $K$ be a normal subgroup of $G$. Is the center $Z(G)$ of $G$ contained in $K$?
4
votes
2answers
161 views

Computing bicharacters of (small) finite groups

I'm trying to find some finite groups with certain properites (hopefully of small order; no more than 100, I suspect), and one of the things I need to look at are all of its bilinear bicharacters: ...
3
votes
3answers
1k views

If $(ab)^3=a^3 b^3$, prove that the group $G$ is abelian. [duplicate]

If in a group $G$, $(ab)^3=a^3 b^3$ for all $a,b\in G$, amd the $3$ does not divide $o(G)$, prove that $G$ is abelian. I interpreted the fact that $3$ does not divide $o(G)$ as saying $(ab)^3\neq e$, ...
2
votes
2answers
158 views

Number of conjugates of $x \in G$ in $H \triangleleft G$

Let $G$ be a finite group, let $H$ be a normal subgroup of prime index, and let $x \in H$ satisfy $C_H(x) < C_G(x)$. If $y \in H$ is conjugate to $x$ in $G$, then $y$ is conjugate to $x$ in $H$. ...
2
votes
2answers
147 views

Splitting of conjugacy class of an element

Let $G$ be a finite Group, H be a subgroup of $G$ of index $2$, and $x\in H$. Denote by $cl_G(x)$ conjugacy class of $x$ in G and by $cl_H(x)$ the conjugacy class of $x$ in $H$. Question is : $(a)$ ...
5
votes
1answer
186 views

If $G'/G''$ and $G''/G'''$ are cyclic then $G''=G'''$

Prove that, if $G'/G''$ and $G''/G'''$ are both cyclic then $G''=G'''$. I was expecting this proof would be similar to the proof of $$G/Z(G)~~~~ \text{is cyclic} \Rightarrow G~~~ \text{is ...
1
vote
0answers
49 views

Understanding a proof by R.C Lyndon and J.L Ullman.

Here in this article I have difficulties understanding the theorem on page 162. Theorem. Let $A, B$ and $C=AB$ be an elements of group $GL_2(\mathbb{Z})$, all with real fixed points. Suppose that ...
3
votes
4answers
282 views

$(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$

Question is to Prove that $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$. Hint : Find two subgroups of order $2$. I somehow feel that a cyclic group can not have two distinct ...
1
vote
1answer
110 views

Groups in an abstract algebra

I have been thinking my brain to find three different examples for a shape S in the plain, such that its group of symmetries is infinite.Also I was asked to draw each shape clearly why its group of ...
3
votes
1answer
146 views

Cardinalities of generating sets for finite groups

Let $G$ be a group. A set $S \subseteq G$ is called a generating set for $G$ if every element of $G$ can be written as a product of elements of $S\cup S^{-1}$. Call $S$ a minimal generating set for ...
2
votes
1answer
184 views

Is the group of positive rationals with multiplication isomorphic to the group of rational with odd denominator with addition

As part of my homework I was given a list of descriptions of groups and I need to determine which pairs are isomorphic. Here are two I am not sure about: The group $(\mathbb{Q}_+,\cdot)$ of positive ...
3
votes
1answer
161 views

An automorphism of order 2 which fixes only the identity

Let $G$ be a group. Assume there is an element $\phi\in\text{Aut}(G)$ such that $\phi(x)=x\Rightarrow x=e$, where $e$ is the identity of $G$, and that $\phi^2$ is the identity automorphism on $G$. I ...
6
votes
1answer
177 views

Simple subgroup of Symmetric Group

I have the following question: Let $n\geq5$, and suppose that $G$ is a simple subgroup of $S_{n+1}$ of index $k$. Show that if $k\leq2n+2$, then $G=A_{n+1}$ or $G$ is isomorphic to $A_n$. I have ...
8
votes
1answer
1k views

Showing that a finite abelian group has a subgroup of order m for each divisor m of n

I have made an attempt to prove that a finite abelian group of order $n$ has a subgroup of order $m$ for every divisor $m$ of $n$. Specifically, I am asked to use a quotient group-induction argument ...
3
votes
1answer
237 views

Abelian subgroups of $\mathrm{GL}(n,\mathbb{Z})$

Does $\mathrm{GL}(n,\mathbb{Z})$ contain torsion-free abelian subgroups that are not isomorphic to $\mathbb{Z}^k$ for some $k$? Said otherwise (as suggested by comment below), does ...
1
vote
1answer
154 views

Power automorphism and abelian groups

First of all I'm not sure if "Power automorphism" is the correct term, so I apologize if it is not. "Let $G$ be an abelian group of order $n$, and $m$ an integer. $f:G\rightarrow G$ s.t. $f(a)=a^m$. ...
1
vote
2answers
65 views

to find the order of the group generated by

is there any way to find the order of the group generated by $x=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ and $y=\begin{pmatrix}0&i\\i&0\end{pmatrix}$ under matrix multiplication ,other ...
2
votes
1answer
47 views

Proof of $a^{2^p}=x^pax^{-p}=a$ $\forall a \in G$

I have to complete this exercise: "Let $G$ be a group with an element of finite order $n>1$ and exactly two conjugacy classes. Prove that $|G|=2$" The author gives some hints: " Prove the ...
1
vote
1answer
92 views

Prove $N$ is a normal subgroup of $G$ is $\forall a,b\in G, (ab)^n=a^n b^n$, and $N=\{x^n|x\in G\}$.

In a group $G$, we have $(ab)^n=a^n b^n$ for all $a,b\in G$ and one particular 'n'. Let $N=\{x\in G| x^n\}$. Prove that $N$ is a normal subgroup of $G$. Proving $N$ is a subgroup is easy. I can't ...
1
vote
0answers
75 views

Isomorphism between quotients of linear groups

Suppose that $n$ is even. Is it true that $$\mathrm{SL}_n (\mathbb{R})/\{ \pm I \} \times \{\pm 1\} \cong \mathrm{GL}_n(\mathbb{R})/{\sim}$$ where $A\sim B$ if and only if $A=aB$ for some $a\in ...