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

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The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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51 views

Order Preserving Isomorphism

If abelian group G has an archimedean order then there is an order preserving isomorphism $\phi$ of G onto a subgroup of $\mathbb{R}$. Here we can say that G is archimedean totally ordered abelian ...
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68 views

A subgroup problem

If $H$ is of finite index in $G$ prove that there is a subgroup $N$ of $G$, contained in $H$, and of finite index in $G$ such that $aNa^{-1}=N$ for all $a \in G$. This problem is from Herstein's ...
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63 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
4
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1answer
35 views

Proving the Thompson Transfer Lemma

Let $G$ be a finite group of even order $n=2^kr$, $T$ a Sylow-$2$ subgroup of $G$, and $M$ an index $2$ subgroup of $T$. I want to show that if $G$ has no subgroup of index $2$, then every element $x$ ...
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1answer
104 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...
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1answer
69 views

a potential application of the ping-pong lemma?

From my understanding, a simple result of the ping-pong lemma would state that if we have a set of linear transformations (matrices) $A_1,\ldots,A_n$ all of the same dimension, then if ...
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32 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
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2answers
64 views

degree of commutativity

What is the exact definition of the degree of commutativity of a $p$-group? When we use notations $d(G)$ and $c(G)$ for other concepts, what is the best notation for degree of commutativity of $G$?
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102 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
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44 views

I.N. Herstein, “Topics in algebra” group theory section 2.8 example 2.8.1

I.N. Herstein, "Topics in algebra" group theory section 2.8 example 2.8.1 it is written that Let $G$ be a finite cyclic group of order $r$, $G=(a)$, $a^r=e$. Suppose $T$ is an automorphism of $G$. If ...
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3answers
51 views

Subgroups of Sufficiently Large Symmetric Groups / Cayley's Theorem explanation

Here's the question: is every finite group a subgroup of a symmetric group of sufficiently large order? More specifically, if a group $G$ has order $n$, then is it true that $G \le S_{n}$? For ...
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1answer
59 views

FC groups with infinite derived subgroup which are not constructed by direct product of finite groups.

A group $G$ is called FC (finite conjugacny) if every conjugacy class $C$ of $G$ has a finite order. It is called FD if the derived subgroup (constructed by commutators) is finite. It is clear that ...
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24 views

For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
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0answers
29 views

Can a compact topological group have the trivial topology?

I want to show that the following are equivalent for a compact topological group $G$: $G$ is the inverse limit of finite groups $G_i$. There's a family $\left\{N_i\right\}$ of open normal subgroups ...
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1answer
46 views

The relation between orders in a group

G is a group and N is a normal subgroup of G.what is the relation between the order of $x$ and $x.N$?
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1answer
472 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
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2answers
139 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
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1answer
55 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
4
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1answer
92 views

groups of order $p^5$ and exponent p

We know that a group of order $p^5$ is an extra special group. How we can show it doesn't have any abelian subgroup of order $p^4$? Also what is the presentation of this group if its exponent is equal ...
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30 views

representation of extraspecial group

Let $H=\langle x,y\rangle$ be a non abelian group of order $p^3$ which has exponent equal to $p$. Also let $G$ be an extraspecial group of order $p^{2r+1}$ exponent $p$. We know that $G$ is isomorphic ...
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51 views

Homology groups of $SL(2,\mathbb Z)$

I am reading Brown's book "Cohomology of Groups" and I can't solve exercise II.7.1.3.: "It's a classical fact that $SL_2(\mathbb Z) \cong \mathbb Z_6 *_{\mathbb Z_2}\mathbb Z_4.$ Use Mayer-Vietoris ...
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2answers
613 views

Intersections of all subgroups is a nontrivial subgroup, so every element has finite order.

I need help to proof this result: "Let G be a group such that the intersection of all its subgroups other than $\{1\}$ is a subgroup different than $\{1\}$. Then all its elements have a finite ...
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2answers
126 views

Decomposability of free product of groups

Let $A$ be a group that can't be expressed as the free product of two nontrivial groups (that is if $A \cong C * D$ then $C$ or $D$ must be trivial). If $T$ is a quotient of $A$ does it have the same ...
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132 views

Infinite groups such that $G/G'$ has odd order.

Can someone give examples of an infinite group $G$ such that $G/G'$ has odd order.
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1answer
135 views

Free Product of two finite groups

The question entails that I should choose two finite groups, then construct a 'biregular' tree, and show that the action of the free product of the two finite groups on the biregular tree will have a ...
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2answers
34 views

order of dihedral

I am learning abstract algebra, and I don't quite understand the order of the symmetry of dihedral. When you look at a squares, I agree that there will be 8 symmetry. But all the operations have cycle ...
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2answers
28 views

multiplication of permutation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, $a=(1\ 3\ 5\ 2)$, $b=(2\ 5\ 6)$, $c=(1\ 6\ 3\ 4)$, $ab=(1\ 3\ 5\ 6)$, ...
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0answers
39 views

an elementary problem on wreath product groups with combinatorial flavor

Embarrassingly, I got stuck in solving the following elementary exercise. Let $G=H\wr \Gamma$ be a wreath product groups, $H,\Gamma$ are countable discrete groups, when $\xi\in\oplus_{\Gamma}H$, then ...
4
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1answer
116 views

Do these definitions make sense?

Letting $G$ be a group and $S(G)$ be all permutations of $G$, define $$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$ It is easy to check that $L(G)$ is a ...
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28 views

How do you compute the inverse of the following permutation?

g = Row#1 (1 2 3 4 5 6) Row #2( 2 3 1 6 5 4) How do you compute g inverse and what is the identity of g?
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1answer
543 views

On the generators of the Modular Group

The modular group is the group $G$ consisting of all linear fractional transformations $\phi$ of the form $$\phi(z)=\frac{az+b}{cz+d}$$ where $a,b,c,d$ are integers and $ad-bc=1$. I have read that $G$ ...
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651 views

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
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Probability of $\alpha\beta\gamma=\gamma\beta\alpha$ for random permutations of a finite set?

Following up on my previous question Probability that two random permutations of an $n$-set commute?, here's a related question for three elements. Q: If $\alpha,\beta,\gamma$ are chosen uniformly ...
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3answers
100 views

Problem from Herstein (Group Theory)

This is the problem from Topics in Algebra by I. N. Herstein. Part of Example No. 2.2.9: Let $G$ be the set of all $2 \times 2$ matrices $ \left( {\begin{array}{cc} a & b \\ c & d \\ ...
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2answers
34 views

Prove that the symmetric group on $n$ letters, $S_n$, has order $n!$.

Here's my proof in which I've used another theorem to prove this one. I want you suggest me another proof without using this theorem, please. Proof: By the theorem Cardinality of set of injections, ...
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3answers
117 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
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1answer
18 views

Isotropy groups of tetrahedron after identifying its sides

If we identify the 4 sides of a regular tetrahedron in $\mathbb{R}^3$ by letting the group of all isometries of the tetrahedron act on it, what would the resulting space look like? The resulting ...
3
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1answer
194 views

In a free group two elements commute if and only if they are powers of a common element

In other words, $uv = vu$ in $F_n$ if and only if $u=w^m$ and $v=w^n$ for some $w\in F_n$. I would like to prove this without making use of Nielsen-Schreier (every subgroup of a free group is free). ...
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2answers
152 views

Group presentations: What's in the kernel of $\phi$?

I have a question about group presentations (in terms of generators and relations). It's been really bugging me for ages. Would really appreciate any thoughts on this. Cheers, Michael You are 'given' ...
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2answers
50 views

How is this subgroup abelian?

Let $G$ be a finite group of order $2n$ such that half of the elements of $G$ are of order $2$ and the other half form a subgroup $H$ of order $n$. Then I know that $H$ is of odd order because for ...
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0answers
36 views

Behavior of groups under extension

We know that an extension of a solvable group by a solvable group is solvable. Similarly we can find other properties of group extensions here Can someone provide a reference to these statements where ...
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1answer
51 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
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1answer
48 views

number of cycles in a permutation

I have given a permutation let 2 3 1 5 4 that is if initially my string is 1 2 3 4 5 the after one permutation it will become 1 2 3 4 5 3 1 2 5 4 that is the number in first position will go ...
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1answer
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How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
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Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
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1answer
48 views

easy direct limit and inverse limit

I am a very beginner in algebra. I want to ask some very simple questions what are these direct limit and inverse limit of groups: 1) $\mathbb{Z} \longrightarrow \mathbb{Z^2} \longrightarrow ...
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20 views

In an infinite cyclic field of non zero units, characteristic $\neq 2$, can an element $-u \neq u$ be expressed as $u^t$ for some finite integer $t$?

For the sake of a proof using contradiction ( to be used somewhere), Lets assume that an infinite cyclic field $F$ of non zero units exists with characteristic $\neq 2$ . In this infinite cyclic field ...
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2answers
26 views

Normal proper subgroup of product of finite simple groups is isomorphic to one of them

I was wondering if anyone can give me a hint/sketch for the following problem, if possible using elementary group theory methods (I am familiar with the material of, say, chapters 1-4 in Rotman). Let ...
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1answer
97 views

Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...