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

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Fundamental Theorem of Finite Abelian Groups: Subgroups

Show that there are two abelian groups of order 108 that have exactly one subgroup of order 3. $$108 = 2^ 2 \times 3 ^ 3$$ Using the fundamental theorem of finite abelian groups, we have ...
7
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1answer
295 views

What makes simple groups so special?

The classification of finite simple groups was one of the most important problem in group theory. But what makes simple groups so interesting and special?
2
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1answer
174 views

If $H \leq G$, $N \lhd G$, $G=HN'$, then $G = H( \gamma _i N)$ for all $i$

If $H \leq G$, $N \lhd G$, $G=HN'$, then $G = H( \gamma _i N)$ for all $i$. Here, $\gamma_iN$ are the terms in the lower central series of $N$, i.e., $\gamma_1 = N$ and $\gamma_{i+1}N = ...
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1answer
61 views

Subgroups of groups of order $2^{a-1}$

The context here is the following exercise Let $m=2^a$ with $a > 2$. Show that $\mathbb{Q}(\theta_m)$ contains exactly three quadratic subfields. By Galois theory, this reduces to the problem ...
2
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2answers
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Cayley Graph of the Symmetry Group of the Triangular Prism

Draw the Cayley Graph of the symmetry group of the triangular Prism. I am having a difficult time with this question. So far I know that the symmetry group has order 12, and also the symmetry ...
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4answers
148 views

Showing a normal subgroup contains a subgroup

Let G be a finite group, $H \le G$, and $N\lhd G$. Suppose $|H|$ and $|G :N|$ are relatively prime. Is it true that $H \le N$? Since $N$ is a normal subgroup, I know that $NH \le G \implies |NH|$ ...
2
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1answer
68 views

Order of an element in the group $(\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times}$

For p a prime and n a positive integer, consider the group of units, $(\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times}$. How can I go about to find the order of $\bar{p}$?
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2answers
140 views

Does taking closure preserve finite index subgroups?

Let $K \leq H$ be two subgroups of a topological group $G$ and suppose that $K$ has finite index in $H$. Does it follow that $\bar{K}$ has finite index in $\bar{H}$ ?
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1answer
204 views

$\operatorname{Hom}(Z_p, Z_p) = Z_p$?

Is $\operatorname{Hom}(\mathbf{Z}_p, \mathbf{Z}_p) = \mathbf{Z}_p$? My proof: $\operatorname{Hom}(\mathbf{Z}_p, \mathbf{Z}_p) = \operatorname{Hom}(\mathbf{Z}_p, \varprojlim\mathbf{Z}/p^n) = ...
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5answers
322 views

A criterion for a group to be abelian

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let and ...
2
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1answer
192 views

Correspondence between normal subgroups of a group and normal subgroups of a quotient groups

Suppose $G$ be a group and $N$ be a proper normal subgroup such that it is not contained in any other proper normal subgroup. Does that mean $G/N$ is simple? Is there one-one correspondence between ...
4
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1answer
169 views

generator of cyclic group

I've been reading this article (http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.0024v2.pdf, page 7, paragraph 2) about a generalized Goursat lemma and in the article the author determines the cyclic ...
5
votes
2answers
221 views

$F_2$ is residually finite, but what are the trivially-intersecting subgroups?

A group $G$ is residually finite if for all $g\in G$ with $g\not=1$ there exists a normal subgroup of finite index, $N_g\lhd_f G$ such that $g\not\in N_g$. Note that $\cap_{g\in G} N_g=1$. It is ...
6
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0answers
181 views

A problem in transfer theory

This is about problem 5B.1 page 157 in Isaacs' "Finite Group Theory" book. This chapter is definitely giving me trouble. The problem reads: Let $G$ be a finite group, $P \in \operatorname{Syl}_p(G)$ ...
2
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0answers
163 views

If $G$ is a finite group and $(ab)^3= a^3b^3$, [duplicate]

Possible Duplicate: Group with an endomorphism that is “almost” abelian is abelian If $G$ is a finite group and $(ab)^3= a^3b^3$, and $3 \nmid o(G)$, then how do I prove that ...
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4answers
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Group of order 15 is abelian

How do I prove group of order 15 is abelian? Is there any general strategy to prove that a group of particular order(composite order) is abelian?
2
votes
2answers
317 views

If a finite simple group acts transitively on a set, is the set finite?

A friend recently asked me if a finite simple group acts transitively on a set, then is the set finite? I want to say yes, since if the action is transitive, then the cardinality of the orbit of any ...
2
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2answers
2k views

Order of product of two elements in a group

Let $G$ be an abelian group. Let $a, b \in G $ and let their order be $m$ and $n$ be respectively. Is it always true that order of $ab$ is $lcm(m,n)$? What if $m$ and $n$ are coprime to each other?
7
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3answers
550 views

Square free finite abelian group is cyclic

How do I show every abelian group whose order is square free is cyclic without using the fundamental theorem of finite abelian groups? I tried something like this Let $|G| = p_1p_2...p_n$ By ...
12
votes
4answers
2k views

Groups of order $pq$ without using Sylow theorems

If $|G| = pq$, $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems?
3
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1answer
280 views

Subgroups of $G$, $|G| = p^n$

If $G$ is a finite group, $|G| = p^n$ , $p$ is a prime, then how do I prove $G$ has a subgroup of order $p^k$, for each $k$, $1 \leq k \leq n$?
2
votes
1answer
100 views

$|G| = p^n$ , $p$ is a prime, and $H$ is subgroup of $G$

If $|G| = p^n$ , $p$ is a prime, and $H \neq G$ is subgroup of $G$ , then how to prove that $N_G(H) \supsetneq H $ ?
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4answers
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Normalizer of the normalizer of the sylow $p$-subgroup

If $P$ is a Sylow $p$-subgroup of $G$, how do I prove that normalizer of the normalizer $P$ is same as the normalizer of $P$ ?
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1answer
183 views

internal direct product groups

Let $G = \{ 3^a 6^b 10^c: a, b, c \in \mathbb{Z}\}$ under multiplication and $H = \{ 3^a 6^b 12^c: a, b,c \in \mathbb{Z}\}$ under multiplication. Prove that $G = \langle3\rangle \times \langle6\rangle ...
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1answer
519 views

Dihedral Group - Internal Direct Product

I have to prove that $D_4$ cannot be the internal direct product of two of its proper subgroups.Please suggest. Since the order of the group is $8$. Internal direct is possible if there exists two ...
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2answers
228 views

Finite/Infinite Coxeter Groups

In the same contest as this we got the following problem: We are given a language with only three letters letters $A,B,C$. Two words are equivalent if they can be transformed from one another ...
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2answers
925 views

Which $p$-groups can be Sylow p-subgroups with trivial intersection?

Every cyclic p-group is a Sylow p-subgroup of a finite group whose distinct Sylow p-subgroups intersect trivially in pairs (and there is at least one pair). For instance, let q be a prime congruent ...
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2answers
255 views

Are all rings groups? [closed]

Also are all groups rings? The book I'm reading is saying that it's not necessary a ring if you have a group. However, that is strange because rings are used to define groups. A ring is group that is ...
2
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2answers
75 views

Do there exist families of groups $G_{s}$ such that $\forall s\in[0,1], |G_{s}|=\mathfrak{c}$

Do there exist families of isomorphism classes of groups $G_{s}$ such that $\forall s\in[0,1], |G_{s}|=\mathfrak{c}$, where $\mathfrak{c}$ is the cardinality of the continuum, and for any ...
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2answers
304 views

A Particular Two-Variable System in a Group

Suppose $a$ and $b$ are elements of a group $G$. If $a^{-1}b^{2}a=b^{3}$ and $b^{-1}a^{2}b=a^{3}$, prove $a=e=b$. I've been trying to prove but still inconclusive. Please prove to me. Thanks very ...
3
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1answer
313 views

Finite Subgroups of general linear group

If $H$ is a finite subgroup of $GL(n,\mathbb{Z})$ then by Minkowskie's theorem, it injects to a subgroup of $GL(n,\mathbb{Z}/p\mathbb{Z})$ under the natural map from $GL(n,\mathbb{Z})$ to ...
4
votes
1answer
106 views

Conjugacy of projective representations

Given characters of the Schur covering group of $G$ of the same degree, how does one tell if the projective representations (as homomorphisms from $G$ into $\operatorname{PGL}$) are conjugate in ...
19
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1answer
306 views

Finite Groups with a subgroup of every possible index

Suppose $G$ is a finite group, with $|G|=n$. Suppose also that for every positive integer $m\mid n$, $G$ has a subgroup of index $m$. Are there any general statements (structural or otherwise) I can ...
2
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1answer
134 views

Projective Tetrahedral Representation

I can embed $A_4$ as a subgroup into $PSL_2(\mathbb{F}_{13})$ (in two different ways in fact). I also have a reduction mod 13 map $$PGL_2(\mathbb{Z}_{13}) \to PGL_2(\mathbb{F}_{13}).$$ My question is: ...
3
votes
1answer
118 views

computing centralizers in the von dyck group $\langle a,b | a^3=b^3=(ab)^3 =1\rangle$

Let $G = \langle a,b | a^3=b^3=(ab)^3 =1\rangle$. I'm trying to compute centralizers in $G$; in particular, I'm interested in the centralizers of $ab$, $ba$, $a^2$, and $b^2$. Does anyone know a good ...
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1answer
67 views

Finding the center of a group with MAGMA

Given a finitely presented group $G = \langle S | R \rangle$, is there a command in MAGMA that computes the center of G ?
8
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2answers
260 views

If $G/Z$ is a product of $p$-groups…

Let $G$ be a finite group. Suppose $Z \le Z(G)$ and $G/Z \cong A\times B$, where $A$ is a $\pi$-group and $B$ is a $\pi'$-group for some set of primes $\pi$. Is it true that $G\cong C \times D$, ...
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3answers
4k views

Prove that the center of a group is a normal subgroup

Let $G$ be a group. We define $H$ where $H$ is the center of/centralizer of $G$: $$H=\{h\in G| \forall g\in G: hg=gh\}.$$ Prove that $H$ is a (normal) subgroup of $G$.
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3answers
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How does one compute the sign of a permutation?

The sign of a permutation $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula $${\rm sgn}(\sigma) ...
2
votes
1answer
79 views

Non conjugate $p$-subgroups of $\mathrm{GL}_n(\mathbb Z)$

It is probably not that difficult but I can't find an example of two non-conjugate $p$-subgroups (of same order) of $\mathrm{GL}_n(\mathbb Z)$ ($n>1$).
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2answers
511 views

Groups of order 12 without Sylow

It is clear that Sylow theorems are an essential tool for the classification of finite groups. I recently read an article by Marcel Wild, The Groups of Order Sixteen Made Easy, where he gives a ...
11
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1answer
692 views

$A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ only simple group of order 360

How can one show, without the use of character theory, that $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ is, up to isomorphism, the only simple group of order 360?
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2answers
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what is exactly the index of a subgroup?

If we have two groups $H$ and $G$ such that $H$ is a subgroup of $G$ we define the index of $H$ in G by the number of all coset of the form $gH$ when $g$ describes $G$ , but I am confused , because ...
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3answers
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Group theory applications along with a solved example

As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and ...
13
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3answers
290 views

How to see that the polynomial $4x^2 - 3x^7$ is a permutation of the elements of $\mathbb{Z}/{11}\mathbb{Z}$

This is from Rotman's Group Theory book, although I don't have the specific reference right now, as the book is with a friend. He asks to show that $\alpha (x) = 4x^2 - 3x^7$ is a permutation of the ...
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1answer
96 views

The permutation group corresponding to translations in three direction on a discrete lattice

What is the name of the permutation group corresponding to all the translation operations in the $3$ directions $x$, $y$ and $z$ (with periodic boundary conditions) of a general rectangular discrete ...
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0answers
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Fundamental Group of Seifert-Fibred Space, as constructed in Hatcher

In Hatcher's notes on 3-Manifolds (available here), he constructs Seifert-fibred spaces in the following way: Let $S$ be some surface, possibly with boundary (let's say with boundary for now). Let ...
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1answer
111 views

Show that the class of a nilpotent group cannot be bounded by a function of the derived length

If $G$ is a nilpotent group with positive class $c$, its derived length is at most $[\mathrm{log}_2c]+1$. This statement can be proved by the inclusion of groups in the derived series and central ...
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3answers
2k views

Right identity and Right inverse implies a group

Let $(G, *)$ be a semi-group. Suppose $ \exists e \in G$ such that $\forall a \in G,\ ae = a$; $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$. How can we prove that $(G,*)$ is ...
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1answer
85 views

Can a group be defined in terms of a relation on a set?

Wikipedia defines a group as "an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element." I keep thinking that there is a ...