The study of symmetry: groups, subgroups, homomorphisms, group actions.

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5
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group-like structure texts.

I was reading Dummit and Foote to be ready for my group theory text, but my teacher seems to be paying special attention to things with less structure than groups, for example monoids, semigroups, and ...
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2answers
57 views

A finite divisible group is trivial

I am having trouble seeing why a finite divisible group is necessarily trivial. Why does this have to be the case?
1
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1answer
47 views

Are the two inverses in the free group same?

Set G is a group, and the set C is contained in G. Set C contains only 2 elements a and b such that they are the inverse of each other.Set F(C) is the free group made by C. Then in F(C), it contains ...
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1answer
35 views

Decomposing $g=xy$ where $\left|g\right|=\left|x\right|{\cdot}\left|y\right|$

Throughout, let us assume we are working with a finite group $G$. The order of an element $g\in G$ is denoted by $\left|g\right|$. It is a standard exercise to prove that if $x, y\in G$ have ...
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39 views

A subgroup of functions under multiplication

Let $G$ be a group of functions from $R$ to $R^*$ under multiplication. Let $H=\{f\in G| f(1)=1\}$. Prove that $H$ is a subgroup of $G$. Can someone make the structure of $G$ more clear possibly with ...
-2
votes
1answer
60 views

Even elements in a cyclic group

Suppose $n$ is an even positive integer and $H$ is a subgroup of $Z_n$. Prove that either every member of $H$ is even or exactly half the members of $H$ are even. Let $n=2m$. I think every subgroup ...
-2
votes
1answer
53 views

Centraliser of a cyclic subgroup

Suppose $a$ belongs to a group $G$ and order of $a$ is $5$. Prove that $C(a)=C(a^3)$, where $C(a)$ is the centralizer of $a$ in $G$. We can show that $C(a)=C(a^4)$ as we always have the identity ...
2
votes
1answer
30 views

Criteria of regularity that a Cayley's Diagram should meet .

As referred in the Visual group theory Book by Nathan Carter- The unofficial definition of a group says that : A group is a collection of actions satisfying the rules: 1. there is a predefined list ...
1
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1answer
32 views

Order of a group of matrices

I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$. So I was thinking on the homomorphism $\det: G \rightarrow ...
1
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2answers
112 views

Find all group homomorphisms from $\mathbb{Z}^n$ to $ \mathbb{Z}^n$

Can we find all group homomorphism from $\mathbb{Z}^n$ to $\mathbb{Z}^n$? For such a map surjective always imply isomorphism (like $\mathbb{Z}$)?
2
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1answer
24 views

Relative orders of an element with respect to a subgroup

There is a theorem in an old monograph: Theorem 1. A pair of subgroups $A$ and $B$ forms a distributive pair if and only if for every element $c$ of $A\vee B$, not in $A$ nor in $B$, its relative ...
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2answers
48 views

Is $<\mathbb Q^+, \times>$ the free abelian group on countably infinitely many generators?

It seems to make sense to me that it should be, with the generators being the set of primes. However, I'm not sure that my intuition is right. Additionally, would this not be contradicted by the fact ...
2
votes
1answer
24 views

Nilpotent and Invertible elements in commutative ring with 1

Let $R$ be a commutative ring with $1$, $S$ a subring also with $1$. Suppose $R\setminus S$ contains a nilpotent element. Prove that $R\setminus S$ also contains an invertible element. Attempt at ...
2
votes
1answer
46 views

what are all finite subgroups of $\mathbb{Z}^n \rtimes \mathbb{Z}_2$?

$\mathbb{Z}^n \rtimes \mathbb{Z}_2$ := $\{(u_1,u_2....,u_n,t), u_iu_j=u_ju_i, tu_jt=u_j^{-1},t^2=1\} $ what are all finite subgroups of $\mathbb{Z}^n \rtimes \mathbb{Z}_2$? (in terms of the ...
1
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0answers
23 views

A question regarding conjugacy classes of central involutions.

An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$. Clearly if an involution is central then its ever conjugate is also ...
2
votes
0answers
48 views

If G is direct product of nonabelian & simple groups. Then prove that Aut(Aut(G)) has only inner automorphisms.

If G is direct product of nonabelian & simple groups. Then prove that Aut(Aut(G)) has only inner automorphisms. Here we can restrict any map f belongs to Aut(Aut(G)) then restrict f to Soc(Aut(G)) ...
2
votes
1answer
27 views

A question on the automorphism of simple graph with distinct eigenvalues of adjacency matrix

Let G be a graph. If G is simple(i.e no loops), and the eigenvalues of adjacency matrix A are distinct, then the automorphism of G is abelian. It seems that the automorphism from G to G itself is only ...
1
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1answer
34 views

$\operatorname{Soc}(\operatorname{Aut}( G))$ is isomorphic to $G$, for $G$ a nonabelian, simple group.

Prove that $\operatorname{Soc}(\operatorname{Aut}(G))$ is isomorphic to $G$, for $G$ a nonabelian, simple group. Here, $\operatorname{Soc}(G)$ is the subgroup generated by all the minimal normal ...
2
votes
1answer
68 views

If a finite set $G$ is closed under an associative product and both cancellation laws hold, then it is a group

Problem Suppose a finite set $G$ is closed under an associative product and that both cancellation law hold in $G$. Prove that $G$ must be a group. Also show by an example that if one just assumed ...
0
votes
1answer
40 views

easy short exact sequence question

Suppose I have have a short exact sequence of finitely generated Abelian groups $0 \longrightarrow G \overset{f}\longrightarrow H \overset{g}\longrightarrow K \longrightarrow 0$. Suppose I have a ...
1
vote
1answer
40 views

Show whether the following groups are cyclic or not…

Here's the full problem: Show whether the groups $G_{10},G_{7}$ are cyclic or not. If so, find their generators. $G_{10}$ and $G_7$ are the sets of invertible elements mod 10 and mod 7. So, I ...
2
votes
2answers
42 views

Problem from Herstein

This is a question from Herstein. I am stuck while trying to solve. Thank you in advance for help. Let G be a group such that the intersection of all its subgroups different from $<e>$ is a ...
0
votes
0answers
43 views

Disprove for any subgroup $H$, we have $N(H)=\{g \in G\mid H\supseteq gHg^{-1}\}$. [duplicate]

Disprove for any subgroup $H$, we have $N(H)=\{g \in G\mid H\supseteq gHg^{-1}\}$, where $N(H)$ is the normalizer of $H$ in a group $G$ i.e. $N(H)=\{g \in G \mid H=gHg^{−1} \}$.
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1answer
20 views

Infinite central series of a group

Is there any definition for an infinite central series? In this link only finite ones are acceptable: http://en.wikipedia.org/wiki/Nilpotent_group#Definition
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2answers
21 views

Torsion elements in integer-modules

In a worryingly short amount of time I've managed to forget almost everything I knew about modules, groups, rings e.t.c. I'm using the definition that an element $m$ of a module $M$ over an integral ...
0
votes
1answer
27 views

Normalizer of a group

Let $ H\leq G $ and $g\in G$. Prove that $N(g^{-1} Hg)=g^{-1}N(H)g$. Here $N(g^{-1}Hg)$ is the normalizer of $g^{-1}Hg$. Can somebody please help me with this problem?
7
votes
1answer
115 views

Showing that group of orientation preserving isometries of Icosahedron is a simple group

Let $G$ denote the group of orientation preserving isometries of Icosahedron. To prove the claim, I have shown that $\nexists \ N \ \triangleleft \ G$ such that $|N|=5.$ $\nexists \ N \ ...
2
votes
2answers
24 views

The set of one-parameter subgroup of the Multiplicative group $G_m$ is Z

Let $G_m= k^{*}=k-{0}$ be the multiplicative group. We know this is an Algebraic group also. How does one prove any algebraic group morphism $G_m \rightarrow G_m$ is of the form $t \mapsto t^{n}$ for ...
2
votes
2answers
114 views

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
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$C_{Aut(Q)}(Inn(Q)) = Inn(Q)$?

Let $Q$ be the quaternion group of order $8$. Write $G = Aut(Q)$ and $N = Inn(Q)$. Show that $C_G(N) = N$ What I tried I know that $C_G(N) = \{g \in G : gN = Ng\}$. I already know that $nN =Nn$ if ...
3
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1answer
94 views

Any subgroup of an abelian group is undistorted.

I need some help with the following math problem. I am studying some notes on Geometric Group Theory and I came across the following problem. Prove: Any subgroup of an abelian group is ...
2
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0answers
32 views

If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$.

If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$. I am reading through some notes on Geometric Group Theory and I came across this side ...
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0answers
45 views

When $H \subsetneq N_G(H)$?

For which subgroups $H$ of a given group $G$ , is it true that $H$ is a proper subgroup of $N_G(H)$ , the normalizer of $H$ ?
4
votes
1answer
35 views

Is the number of associative $n$-ary algebraic operations on a finite set with 2 cardinality always 8?

We know that if $n = 2$ then the operation is called a binary operation. $ \circ $ on set $X$ is a function $\circ : X \times X \rightarrow X$. And the number of all associative binary operation on a ...
2
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1answer
27 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
6
votes
2answers
538 views

Prove that a group with certain property is abelian

Let $G$ be a group with the following property: If $a, b,$ and $c$ belong to $G$ and $ab=ca$,then $b=c$. Prove that $G$ is abelian. I think I have the answer for the finite $G$ case. Then, for a ...
3
votes
1answer
57 views

Generating set of a group

A subset $S$ of a group $G$ is said to be a generating set for $G$ if all elements of G can be expressed as the FINITE product of elements in S and their inverses. Why is it necessary to take only ...
2
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0answers
27 views

About Chernikov Groups

A Chernikov group is a group $G$ that has a normal subgroup $N$ such that $G/N$ is finite and $N$ is a direct product of finitely many quasicyclic groups. My question is: If $G_1$ and $G_2$ are ...
2
votes
2answers
87 views

$\forall a \in \mathbb{Z}, \quad a^{13} = a \bmod 35$

How does one prove that $\forall a \in \mathbb{Z}, \quad a^{13} = a \bmod 35$? I would recall that $a^p = a \bmod p$ if $p$ does not divide $a$, but $13 \neq 35$ and besides the statement should ...
2
votes
2answers
70 views

Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
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0answers
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Automorphisms of $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$

Let $p$ a prime and $n$ a positive integer, What are the outer automorphisms of the finite linear group $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$? do we know a complete list of them? Is there any thing on ...
6
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3answers
200 views

Regarding the composition of permutations…

So I have the permutations: $$\pi=\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8 ...
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votes
1answer
27 views

Stablizer of action in kernel of character

Let $N$ be a positive integer. Let $G=(\mathbb{Z}/N\mathbb{Z})^\times$. Set $$G^*=\{\chi:G\to \mathbb{C}\mid \chi \text{ is homomorphism}\}.$$ Let $X$ be set, assume that $G$ acts on $X$. For $x \in ...
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0answers
25 views

Question on extension of cocycles

Given a countable discrete group $G$ and suppose $G$ acts on a compact metrizable abelian group $Y$ with normalized Haar measure $\mu$, measure preserving, let $\mathbb{T}$ denotes the unite circle. ...
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1answer
40 views

On special type of p-group of class two

Let $G$ be a finite $p$-group of class two and $T$ be the minimal set of generators of $G$. Also let $G=\langle a, b, x_{1},..., x_{n}\rangle$ such that $|a|=|b|=p^n$, $a,b\in T$, $x_{1},..., ...
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1answer
37 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that ...
3
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generators of the automorphism group of $\bigoplus_{i=1}^\infty \mathbb{Z}$

in some helpful comments recently Dustan Levenstein directed my attention to the automorphism group: $$ A\left(\bigoplus_{i=1}^\infty \mathbb{Z}\right) $$ i think the direct sum means that any ...
4
votes
1answer
124 views

Prove that intersection of finite index subgroups has finite index.

I'm trying this problem from Herstein: Q) If G is a group and H, K are two subgroups of finite index in G, prove that H $\cap$ K is of finite index in G. Can you find an upper bound for the index ...
8
votes
2answers
65 views

$A_n \times \mathbb{Z} /2 \mathbb{Z} \ \ncong \ S_n$ for $n \geq 3$

I try to show that $A_n \times \mathbb{Z} /2 \mathbb{Z} \ \ncong \ S_n$ for $n \geq 3$. It is not hard to show the statement for $n=3$. We have $$ A_3 \times \mathbb{Z} /2 \mathbb{Z} \ \cong \ ...
2
votes
2answers
45 views

Show that there is just one subgroup $H \subset S_4$ such that $[S_4:H] = 2$

I had to show that $S_4$ has one subgroup of index $2$. Below, you'll find what I tried to do so Of course, $A_4$ is a subgroup index $2$. To show that there is not another subgroup with the same ...