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

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12
votes
7answers
2k views

Applications of the wreath product?

We recently went through the wreath product in my group theory class, but the definition still seems a bit unmotivated to me. The two reasons I can see for it are 1) it allows us to construct new ...
1
vote
2answers
178 views

Wreath Product of $\mathbf{Z}_{p^n}$ and $\mathbf{Z}_{p^m}$

Is this true that: "The order of wreath product of $\mathbf{Z}_{p^n}$ and $\mathbf{Z}_{p^m}$ is $p^{m+n}$" ?
6
votes
1answer
190 views

Isoperimetric inequalities of a group

How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being ...
4
votes
3answers
5k views

What is $\gcd(0,a)$, where a is a positive integer?

I have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is ...
2
votes
1answer
432 views

Center of G/Z(G)

How does one find the center of the group G/Z(G)?
4
votes
1answer
428 views

Finite abelian groups and isomorphism

Let $G$ and $H$ be finite abelian groups such that $G \times G \cong H \times H$ Then $G \cong H$ I was going to just write the hypothesis as $G^2 \cong H^2$ and take square roots on both sides, but ...
3
votes
1answer
126 views

groups and checking group properties

I am wondering if the nonzero rationals with operation $x\circ y = xy/7$ a group. I think it may be and I found the identity to be 7 and found inverses for each element. Mark
3
votes
2answers
116 views

Question about characteristic subgroups

If N is characteristic in a group G and if we have that K/N is characteristic in G/N why do we have that K is characteristic in G.
7
votes
2answers
1k views

Number of automorphisms of a direct product of two cyclic $p$-groups

Suppose I have $G = Z_{p^m} \times Z_{p^n}$ for $m, n$ distinct natural numbers and $p$ a prime. Is there a combinatorial way to determine the number of automorphisms of $G$?
0
votes
1answer
464 views

p-Groups with every proper subgroup cyclic

The quaternion group is non-abelian p-group with every proper subgroup cyclic. In general, if $G$ is a non-abelian $p$-group with every proper subgroup cyclic, is it necessarily generalized ...
4
votes
1answer
107 views

Kernel of $\phi: A * B \to A \times B$ is free

I have come across this question in a course about Cayley graphs recently. I don't really have a clue how to answer this question. Here it is. Consider two groups $A,B$. Let $K< A*B$ be the kernel ...
7
votes
3answers
449 views

Embedding of finite groups

It is well known that any finite group can be embedded in Symmetric group $S_n$, $GL(n,q)$ ($q=p^m$) for some $m,n,q\in \mathbb{N}$. Can we embed any finite group in $A_n$, or $SL(n,q)$ for some ...
0
votes
2answers
966 views

What is the difference between the order of a group and the order of the elements of the group

I know the order of a group is the size of the group, ie the number of elements. But what does it mean for an element of that group to have order? Also, what are the precise definitions for 1) ...
9
votes
2answers
820 views

$A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
4
votes
2answers
227 views

Defining sets and union as a group

I came across an exercise in this book where the question was to define a collection of sets and the union operator as a group. The two parts of the question were to (1) find the identity element and ...
6
votes
3answers
387 views

Alternative “functorial” proof of Nielsen-Schreier?

There are two proofs of Nielsen-Schreier that I know of. The theorem states that every subgroup of a free group is free. The first proof uses topology and covering space theory and is rather elegant. ...
2
votes
1answer
706 views

Automorphisms of projective special linear group

How to prove that $Aut(PSL(2,7))=PGL(2,7)$? Is this result extendible to $PSL(n,q)$ where $q=p^n$?
4
votes
2answers
528 views

Questions on Möbius function

I was reading a paper, related to group theory. I came across two doubts (may be simple, but puzzling me): Let $G$ be a finite group and consider the lattice of subgroups of $G$. On this lattice, ...
1
vote
2answers
195 views

Parameterization of modular group

Is there a parameterization by integers of the modular group SL(2,Z)? What I mean is, some expression for a matrix in terms of several variables (a,b,c,...) such that for each n-tuple of integers ...
5
votes
1answer
277 views

Braid groups and representations

I was wondering if $\mathbb{Z} \wr S_n$, where $\mathbb{Z}$ is the usual group of integers, $S_n$ the symmetric group on n elements and $\wr$ the wreath product of two groups, contains the braid group ...
7
votes
2answers
736 views

Why is the generalized quaternion group $Q_n$ not a semi-direct product?

Why is the generalized quaternion group $Q_n$ not a semidirect product?
2
votes
1answer
485 views

Homomorphisms and normal subgroups proof

Let $f$ be a homomorphism mapping $G$ to $J$ groups. Assume that $J$ is abelian. Prove that if $H$ is a subgroup of $G$ and if $\mathrm{ker}(f)$ is a subset of $H$, then $H$ is normal in $G$. I ...
4
votes
3answers
160 views

Inclusion of subgroups implies the group is cyclic

Let $G$ be a finite group such that for any two subgroups $H_{1}$ and $H_{2}$ of $G$ we have $H_{1} \subseteq H_{2}$ or $H_{2} \subseteq H_{1}$. Why this implies $G$ is a cyclic group? Ah. I think ...
26
votes
4answers
2k views

Center-commutator duality

I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6: Any subgroup of $G$ which contains $[G,G]$ is normal in $G$. He says this as ...
0
votes
2answers
134 views

Question about indices of subgroups

Suppose $H$,$K$ are normal subgroups of a group $G$. We don't assume $G$ is finite. Furthermore assume $[G:H]$ and $[G:K]$ are both finite. Then it follows that $[G: H \cap K]$ is divisible by ...
1
vote
0answers
294 views

Mobius function for finite groups

For finite group $G$, Mobius function is defined on the lattice of subgroups of $G$ as: $\sum_{H\geq K} {\mu (H)}=\delta_{K,G}$. From this, $\mu (H)$ can be obtained by induction argument. 1. Can we ...
3
votes
2answers
3k views

Order of a particular given permutation = LCM(order of all disjoint cycles) ?

If I write (1 3 4 6)(2 3 4)(4 6 1) as a product of disjoint cycles, I get (164)(23), is it true that order of (1 3 4 6)(2 3 4)(4 6 1) is lcm(2,3)=6 (just the order of its disjoint cycles)?
5
votes
3answers
484 views

Other ways to deduce Cyclicity of the Units of certain groups?

The group of units of the rings $\text{GF}(p^r)$ and $\mathbb{Z}/p^r\mathbb{Z}$ are both cyclic (except for the exception of prime powers are not cyclic when $p=2$ and $r\ge 3$). This is a strong ...
3
votes
3answers
143 views

number of arrangements

In a standard deck, there are 13 cards in each suit,such as hearts. For simplicity we'll number them from 1 to 13. This question is about these 13 cards. Start with the cards in the usual order. ...
8
votes
3answers
371 views

Is $SO_2$ an amenable group?

In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows: [amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined ...
4
votes
1answer
147 views

Well known proofs that $\mathrm{psl}_2(\mathbf Z) \cong C_2 * C_3$

I've seen one proof of $\mathrm{psl}_2(\mathbf Z) \cong C_2 * C_3$ based on the Ping-pong lemma But what I'm looking for is how this result was historically obtained first. I've checked Coxeter ...
0
votes
1answer
117 views

Galois theory help needed for field extensions

Let $K$ be any field and $s$ an indeterminate. Then $K(s)$ is a field extension of $K (s^n )$. Prove that $[K (s):K (s^n )]=n$. Hence show that the minimum polynomial of $s$ over $K(s^n)$ ...
3
votes
1answer
510 views

Question regarding galois theory : finding a fixed field of a group generated by a C-automorphism

Let $L=\mathbb{C}(t)$ (where $t$ is an indeterminate). Let $\phi\colon L\to L$ be the $\mathbb{C}$-automorphism of $L$ given by $$\phi(t) = \frac{3t-2}{4t-3},$$ and let $G$ be the group ...
2
votes
1answer
142 views

Sub fields of $\mathbb{C}$ that are splitting fields over $\mathbb{Q}$

Hi I really need help in how to determine sub fields $L$ of $\mathbb{C}$ over $\mathbb{Q}$ and finding $[L:\mathbb{Q}]$. The question is: Find subfields $L$ of $\mathbb{C}$ that are splitting fields ...
12
votes
12answers
6k views

Introductory Group theory textbook

What textbook is good for introductory group theory?
2
votes
1answer
261 views

Direct sum with isomorphic groups

A follow up to a previous question of mine. I thought it was true, but according to my book $G \simeq H \oplus K$, with $G \simeq H$ does not imply $K=0$ Is there a simple counter example? In what ...
56
votes
4answers
2k views

How is a group made up of simple groups?

I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers. I've recently started ...
5
votes
2answers
205 views

A question on $\operatorname{GL}_2(\mathbb R)$

I know that all finite subgroups of $\operatorname{SL}_2(\mathbb R)$ are cyclic by standard averaging argument. They are all conjugate to some finite subgroup of $\operatorname{SO}_2(\mathbb R)$ and ...
5
votes
2answers
627 views

Showing groups of order $p^{k}(p+1)$ are not simple, p prime

I want to show that there are no simple groups of order $p^{k}(p+1)$ where $k>0$ and $p$ is a prime number. So suppose there is such a group. Then if we let $n_{p}$ denote the number of $p$-Sylow ...
10
votes
2answers
477 views

Working out a Group Presentation

If you have a group, (say you have group table or any other information), is there an algorithm to find the group presentation? What is the general way of finding presentation of a group?
2
votes
2answers
162 views

number of elements in $S_{13}$ conjugate to both (12)(34) and (123)(45)

Two permutations are conjugate of each other if and only if they have the same cycle structure. If I want to find the number of elemnts in $S_{13}$ that are both conjugate to (123)(45) and ...
4
votes
1answer
255 views

Getting generators of graphs automorphism group

Suppose I have a graph like this and a list of its automorphisms. How do I go about getting a set of generators for this group?
12
votes
3answers
5k views

What are useful tricks for determining whether groups are isomorphic?

In general, it is not too hard to find isomorphisms between two groups when their order is relatively low. However, as their orders grow, it becomes increasingly irritating to write down their entire ...
13
votes
4answers
1k views

Simple Group Theory Question Regarding Sylow Theorems

It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least $n_p(p^n-1)$ elements, where $n_p$ is the number ...
3
votes
2answers
612 views

Amalgamated Free Product

Suppose $H$ is embedded in $G$ and $H'$ is isomorphic to $H$ and embedded in $G'$. Then we can simultaneously embed $H$, $H'$, $G$ and $G'$ into a single object (the amalgamated free ...
13
votes
1answer
631 views

Rubik's cube interesting questions?

The upper bound for the number of moves required to solve a regular Rubik's cube has been shown to be 20. Two questions come to mind: Does this result have more general significance? What are the ...
4
votes
2answers
364 views

Defining representations; representations and semidirect products

Let $G$ be a group, $X$ a set. Defining an action $G\times X \to X$ is the same as defining a group morphism $\rho: G\to Sym(X)$, through the formula $g\cdot - = \rho(g)$. The morphism $\rho$ is ...
4
votes
2answers
627 views

Show that if $G$ is a finite nilpotent group, then every Sylow subgroup is normal in $G$

Thank you~ Show that if $G$ is a finite nilpotent group, then every Sylow subgroup is normal in $G$. I know that the normalizer of any proper subgroup of a nilpotent group contains this subgroup ...
14
votes
2answers
1k views

Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A ...
4
votes
1answer
136 views

The natural inclusion of an infinite abelian group $G$ into $\widehat{\widehat{G}}$

I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into ...