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.

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How to write the commutator subgroup in terms of the generators of the group?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. The commutator subgroup of $G$ is the group generated by $\{[a,b]\ |\ a,b\in G\}$ and is denoted by $[G,G]$, where $[a,b]=aba^{-1}b^{-1}...
2
votes
3answers
91 views

If $G/N\cong H$ then $G=NH$?

I am curious if $G/N\cong H$ then $G=NH$? ($N$ is a normal subgroup of $G$, and $H$ is a subgroup of $G$.) With this setup, we get that $NH$ is a subgroup of $G$ so $NH\subset G$. I am not sure ...
3
votes
1answer
36 views

Two equivalent conditions proof (related to semiproduct of groups)

Let $G$ be a group, and let $N$ be a normal subgroup of $G$, $H$ any subgroup of $G$. I wish to prove the equivalence of (i) $G$ is the product of subgroups, $G=NH$, where $N\cap H=\{e\}$. (ii) ...
0
votes
1answer
41 views

solve a specific word problem in free groups

Let $F_2=\langle a, b\rangle$ be the non-abelian free group with two generators and $e$ is the neutral element in $F_2$. Given $g\in F_2, k\geq 2$ an integer. I want to know how to solve the word ...
0
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0answers
25 views

Sylow tower theorem involving supersolvable groups

I just want to find out if anyone has a reference to the result that states that if $G$ is a finite supersolvable group then it has a normal Sylow subgroup.
3
votes
1answer
25 views

Prove that L(x) = (x - 1)/p is a discrete logarithm function in a group.

I have the following problem:$$$$ Let $p$ be a prime number and $G$ be a set of all $x\in \mathbb{Z}_{p^2}$, such that $x \equiv 1 \pmod{p}$. Prove that: $G$ is a multiplicative group (regarding ...
0
votes
1answer
44 views

soft question- Kenneth's Brown notation at “Cohomology of finite Groups”

As the title indicates, my question has to do with something rather simple. So, in Kenneth's Brown book "Cohomology of finite groups" at pg.84-85 and in particular Theorem 10.3 and Proposition 10.4, ...
0
votes
1answer
25 views

Reference for results p-adic integers Z_p as abelian group

I have two facts I want to use in my thesis about $\mathbb{Z}_p$. To be precise: automorphism group is $\mathbb{Z}_p \times \mathbb{Z}/(p-1)\mathbb{Z}$, except for 2, and that any subgroup with finite ...
1
vote
0answers
23 views

Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
1
vote
1answer
49 views

Product with a normal subgroup

If $H \unlhd K \unlhd G$ and $P\unlhd G$ then does necessarily $HP \unlhd KP$? I can see this is true using the correspondence theorem since $HP/P \unlhd KP/P$ I want to try direct and prove it ...
2
votes
0answers
65 views

Commensurability of two groups

If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
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vote
1answer
26 views

Supersolvable and pronormal subgroups

Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....
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votes
0answers
49 views

Prove that $G = \{\cos x + t \sin x: x\in\mathbb{R}\}$ is an abelian group under multiplication

where $x$ is the angle Closure is easy - Since $x$ is a real number, its $\cos$ component and $\sin$ component will be a real number. Associative property - I guess it means that $(\cos x + t\sin ...
1
vote
0answers
12 views

Polar of orthogonal set invariant under group action

I just ask the following question: Set invariant under group action Furthermore, How to prove the green part Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324) Let $$O(n)=...
8
votes
2answers
107 views

Making sense of the commutator

For a group $G$, the commutator of two elements is defined as $[a,b]=aba^{-1}b^{-1}$, and is usually said to measure the extent to which the elements $a$ and $b$ fail to commute. I'm having some ...
2
votes
0answers
37 views

Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. ...
1
vote
1answer
26 views

If $d\mid n$, element of $(\mathbb{Z}/n\mathbb{Z})^*$ of highest order is also element of $(\mathbb{Z}/d\mathbb{Z})^*$ of highest order

I am not entirely sure if the following lemma is true, but after running a code to check, for smaller values of $n$, it holds. Can someone outline the proof or give a counterexample for it. Lemma: ...
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vote
0answers
30 views

Combinatorial property of sets

Is the following true? For every $\varepsilon>0$ there is a finite subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that $$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...
2
votes
2answers
21 views

Proof that Möbius transformations are group under composition - finding inverse element

The task given in my textbook was to find which algebraic structure is $(X, *)$, where $X$ is set of Möbius transformations $x\rightarrow y=\frac{ax+b}{cx+d}$ in $\mathbb R$ and $*$ is composition. I ...
3
votes
1answer
57 views

How to determine the order of the group $\langle a,b,c |a^2=b^2=c^2=(ab)^2=(bc)^4=(ca)^4=1 \rangle$?

How to determine the order of the group $\langle a,b,c |a^2=b^2=c^2=(ab)^2=(bc)^4=(ca)^4=1 \rangle$ ? I have almost no idea how to go on about this . Please help . Thanks in advance
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vote
0answers
23 views

Is the convolution algebra a *-algebra?

Let $G$ be a finite abelian group with $n$ elements. Consider the convolution algebra $C^*(G) \subset l^2(G)$, with multiplication: $$(a * b )(g) = \frac{1}{n}\sum_{x\in G} a(x)b(g-x)$$ Is there a ...
1
vote
4answers
109 views

Infinite groups with all elements of order 2?

If G is a group such that $a^2 =e$ for all $a \in G$, where $e$ is the identity element in $G$, then $G$ is finite. This question can be proved false if we can get a group of infinite order with ...
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votes
0answers
26 views

prime-power order subgroups of a quotient group

Let $G$ be a finite group. Suppose that $M$ is normal $p$-subgroup of $G$. Then every prime-power order subgroup of $G/M$ is either a $p$-subgroup or a $q$-subgroup of the form $QM/M$ for some $q$-...
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votes
0answers
42 views

Problem in finding a general rule for determining $o(ab)$ if $o(a)$ and $o(b)$ are known. [duplicate]

Is there any way to determine $o(ab)$ if $o(a)$ and $o(b)$ are known, where $a$, $b$ belong to a group $G$. @Derek Holt I have proved that $o(ab) = o(ba)$.Which is very easy.Let us assume that $o(ab)...
1
vote
1answer
49 views

kernel of product of group homomorphisms

Let $f,g:A \to B$ be group homomorphisms, with $B$ abelian. Then $f\cdot g$ is also a group homomorphism. What can I say about $\ker(f \cdot g)$ in terms of $\ker(f)$ and $\ker(g)$?
1
vote
1answer
42 views

Existence of surjective map / homomorphism from an infinite group onto its symmetric group

From Cayley's theorem , we know that any group $G$ can be embedded in its permutation group $S(G)$ ; I would like to ask , If $G$ be an infinite group , then does there exist a surjection from $G$ ...
0
votes
0answers
55 views

Lie algebra equivalent definition

I am reading the paper "Affine projections of polynomials" by Neeraj Kayal. I need a clarification regarding the equivalence of two definitions of lie algebra : Let $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
8
votes
1answer
149 views

Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
0
votes
1answer
44 views

A block in understadning a proof of Sylow's Theorem [duplicate]

If $G$ is a finite group of order $n$ and $p$ is a prime divisor $|G|$, then according to one of Sylow's theorems $G$ has a subgroup of order $p^m$, where $p^m$ is the highest power of $p$ which is a ...
2
votes
0answers
35 views

Group action with two normal subgroups which induce same block system

So awhile back I asked this question here on stack exchange: Normal subgroup $H$ of $G$ with same orbits of action on $X$. At the time I wasn't quite sure what I was really wanting to know about ...
4
votes
2answers
89 views

group theory, cardinality of a quotient group

Let $H$ be any finite abelian group. Define $H^n=\lbrace x^n \text{ }|\text{ }x\in H\rbrace$. It is easy to see that $H^n$ is a subgroup of $H$. It is also easy to see that $H^n\subseteq H^d$ if $d\...
2
votes
1answer
27 views

Centralizer of a cyclic sylow $p$-subgroup

Let $G$ be a group where $|G|=p^nm$ and $p$ is the smallest prime dividing $|G|$. Suppose that $P \in \operatorname{Syl}_p(G)$ is cylic. Then $C_G(P)=N_G(P)$ Now I know $|N_G(P)/C_G(P)|$ must divide $...
5
votes
1answer
59 views

Simple characterization of integers among abelian groups

This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...
4
votes
2answers
50 views

If $G$ a finite $p$-group s.t $G/[G;G]$ cyclic then $G$ abelian.

Let $G$ be a finite $p$-group and $[G,G]$ its commutator sub-group, I need to show that if the group quotient $G/[G,G]$ is cyclic then $G$ is an abelian group. My attempt is to let $g\in G$ s.t $g$ ...
23
votes
4answers
2k views

Algebraic Structures that do not respect isomorphism

One of the first things a student learn in Algebra is isomorphism, and it seems many objects in algebra are defined up to isomorphism. It then comes as a mild shock (at least to me) that quotient ...
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vote
0answers
35 views

Converse of Lagrange theorem does not hold on $S_n$ for $n\geq 4$. [closed]

I already know that converse of Lagrange theorem does not hold on $A_n$. If converse of Lagrange theorem does hold on $A_n$ , Some group of $A_n$ has all 3cycles and it is whole group. Could you ...
10
votes
4answers
158 views

Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial

The group of rational numbers $(\mathbb Q,+)$ has an interesting property , that the intersection of any two non-trivial subgroups of this group is non-trivial . Let us call this property the " non-...
1
vote
2answers
57 views

Are the groups $\mathbb R/ \mathbb Z $ and $ \mathbb R^2 / (\mathbb Z \times \{0\} )$ isomorphic?

Is it true that as groups , $\mathbb R/ \mathbb Z \cong \mathbb R^2 / (\mathbb Z \times \{0\} )$ ? I only know that $\mathbb R \cong \mathbb R^2$ (as groups ) but I can see no way to decide whether ...
2
votes
2answers
88 views

Are the groups $\mathbb C^* \times \mathbb R^*$ and $\mathbb R^* \times \mathbb R^*$ isomorphic ?

Consider the groups $\mathbb R^* , \mathbb C^*$ under multiplication , I know that they are not isomorphic ( as one of them is divisible but the other is not ) , my question is : Are the groups $\...
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votes
1answer
36 views

Order of automorphism divides order of element

Question: Let $a$ belong to a group $G$ and let $\left | a \right |$ be finite. Let $\phi_{a}$ be the automorphism of $G$ given by $\left ( x \right )\phi_{a}=axa^{-1}.$ Show that $\left | \...
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votes
1answer
23 views

Why is the order of an element in a cyclic group a factor of the order of the cyclic group it is in?

If G has an element a of order k, then the group generated by a consists of {e, a, a^2, ..., a^(k-1)} which are all distinct elements of the group generated by a whose orders are a factor of k. I don'...
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vote
0answers
29 views

Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
2
votes
2answers
53 views

Studying $\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})$ where $G$ is finite.

Let $G$ be a finite order group. Then we can write $|G| = 2^n(2m+1)$ for some non-negative integers $n$ and $m$. I'm trying to show that $\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})$ is an abelian ...
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0answers
25 views

Doubly transitive and solvable

I wonder if there are subgroup of $S_n$ that are solvable and doubly transitive for $n\geq 5$. Can someone find an example or prove that they don't exist?
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votes
1answer
56 views

Automorphism of a finite cylic $p$-group

Let $G$ be a finite group. Suppose that $A$ is a cylic $p$-subgroup and $A\unlhd N_G(Q)$ for some subgroup $Q$ of $G$. Let $x \in N_G(Q)$ be an $p'$-order element. Then $x$ induces an automorphism of ...
0
votes
1answer
29 views

Sylow-p-group of matrices group over finite field.

Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $ U $ on the diagonal. Find a Sylow-p-...
0
votes
1answer
32 views

Let $G$ a group of order $n$. If $k$ divide $n$ in a certain sense, is there an element of order $k$.

Let $G$ a group of order $n$. I was wondering that if $k$ divide $n$ in a certain sense, then there an element of order $k$. I explain. Let $g$ an element of order $m$. We know by Lagrange that $m\mid ...
0
votes
2answers
101 views

Groups and rings of order $p^2$.

Up to isomorphism there are exactly two abelian groups of order $p^2$. there are exactly two groups of order $p^2$. there are exactly two commutative rings of order $p^2$. there is exactly one ...
0
votes
3answers
25 views

Show that the two maps are equivalent

Question: Let G be a group and let $g \in G$. If $z \in Z\left ( G \right )$, show that the inner automorphism induced by g is the same as the inner automorphism induced by $zg$. That is show that,...
1
vote
0answers
32 views

subnormal subgroup of a $p$-group

I could not reason with this statement of a proof I am currently looking at. Let $G$ be a finite $p$-group with $Q\leq G$ and suppose that $H$ is a cyclic subgroup of $Q$ then $H$ is subnormal in $...