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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Computing $|Aut(G)|$ of a given abelian group

I had compute $|Aut(G)|$ of a given abelian group. Now using the fact $(|G_1|,|G_2|)=1$ problem boils down to compute $|Aut(\prod_{i} \mathbb{Z}_{p^{a_i}})|$ for a prime $p$. Now here I'm stuck. For ...
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Lattice of Subgroups and Automorphisms

So I have a rather interesting question that came up in some independent research I have been doing on finite groups of small order. I was looking at their (full) subgroup lattices, which included the ...
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20 views

Shuffles vs direct sums of permutations

A $(p,q)$-shuffle is a permutation of $p+q$ things that preserves the internal order of the first $p$ things and of the last $q$ things. As remarked on wikipedia, since a $(p,q)$-shuffle is uniquely ...
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Prove that there exists $b\in G$ such that $h(x)=xb^{-1} $ for all $x\in G$.

I'm trying to do every problem in my book, but I got stuck on this problem, which seems like it should be easy. Notation: $A(G)$ denotes the group of all permutations of the set $G$. The map $\...
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2answers
237 views

A question about Nielsen–Schreier theorem

I have a question about this well-known theorem about free groups by focusing on the proof stated by D. L. Johnson in his book "Presentations of groups ": Theorem (Nielsen-Schreier): Let $F$ be a ...
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1answer
475 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
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38 views

The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
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1answer
505 views

Subgroup structure of dihedral groups

I am currently looking into structure of dihedral groups; I am interested in their subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them ...
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1answer
19 views

Counting number of cosets

Let $G = \big(\mathbb{Z}/n\mathbb{Z})^*$, that is the multiplicative group modulo $n$. For some $d$ coprime to $n$, let $H$ be a subgroup of $G$ generated by $d$. As $G$ is abelian, $H$ is normal in $...
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(B,N) pair and Steinberg idempotent

Let $q=p^f$ where $p$ is prime and $G$ be a finite group with a $(B,N)−$pair ($T=B\cap N$ and $W=N/T$), and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Define $$e=\dfrac{1}{[G:U]}\...
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60 views

Showing: if G acts on A by conjugation then the stabilizer of A in G is the Normalizer of A in G.

This is a theorem from Dummit & Foote text- The number of conjugates of a subset $ A$ in a group $G$ is the index of the normalizer of $A$,$\vert G:N_G(A) \vert$. The highlighted text is a ...
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27 views

Group endomorphisms of simple abelian groups which do not commute by composition. [on hold]

What is an example of group homomorphisms $f,g: M \to M$ where $M$ is a simple abelian group such that $f\circ g \ne g\circ f$ ?
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61 views

Condition that for a given set of numbers and given divisor all finite sums from this set contain all possible remainders

Given $q \in \mathbb{N}$ and ${a_1, a_2, ...}$ where each $a_j \in \mathbb{N} \cup{\{0\}}$ define $A_p=$ {set of all finite sums of $\{a_1 ... a_p\}$ such that each $a_j$ will appear either $1$ or $0$ ...
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+100

Show that $ \mathcal{D}_H:=\bigcup_{g_i\in[H\backslash G]} g_i\cdot \mathcal{D} $ is a fundamental domain

Let $G$ be a group which acts on the set $X$. Consider a subgroup $H$ of $G$ which acts on $X$ by the restriction of the action of $G$ on $X$. Let $[H\backslash G]:=\{g_i\ \ :\ \ \exists!\mathcal{O}\...
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3answers
37 views

Why is a linear transformation expressed using its transpose?

If $A$ is an invertible matrix with entries from $\mathbb{R}$, what is the reasoning behind defining an invertible linear transformation $f_A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ as $f_A=xA^t$, ...
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1answer
24 views

(B,N) pair and normal subgroup

I am trying to prove the following: Let $G$ be a finite group with a $(B,N)-$pair and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Let $\widetilde{G}\triangleleft G$ such that $U\le \...
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23 views

When is Cartier dual of a finite group etale?

I am trying to solve the following exercise from Waterhouse: Introduction to affine group schemes (Chapter 6, Ex. 12 on page 53) without any success. Let $char(k)=p >0$ and let $G$ be an abelian ...
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10 views

Tannaka Krein duality for finite groups, explicit

Tannaka-Krein duality theory says that the natural mapping $G\rightarrow Aut^{\otimes}(F)$ (see http://mathoverflow.net/questions/155743/can-one-explain-tannaka-krein-duality-for-a-finite-group-to-a-...
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3answers
728 views

Two dimensional complex group representations

Michael Artin's Algebra, chapter 10 (both unstarred, and complex representations) M.8 Prove that a finite simple group that is not of prime order has no nontrivial representation of dimension 2. ...
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How many elements in $S_{8}$ are conjugate with $(12)(345)$?

How many elements in $S_{8}$ are conjugate with $(12)(345)$? My reasoning is as follows: Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the ...
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4answers
90 views

Permutation in group theory [on hold]

I am confuse how to proceed for the following question. Can you please help me. Thanks in advance! For a permutation $\pi$ of $\{1,\cdots,n\}$, one say that $k$ is a fixed point of $\pi$ if and only ...
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In $D_{33}$ how do I find out number of elements of each order?

In $D_{33}$ i.e diehedral group of order 66. How do I find out number of elements of each order? The only idea I have is that possible order of any element can be 1,2,3,6,11,33,66. Now 1 is only for ...
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Result of a Primitive Solvable Group

Suppose that $G$ is a primitive solvable group. Then $F(G) = O_p(G)$ for some prime $p$ Clearly, $O_p(G) \leq F(G)$. Any hints on proving the other inclusion?
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rotating a point using a previously rotated one

I want to rotate a shape in an n dimensional space (n>3) around (about) the origin. knowing the outcome of rotation on a point like A, which is A', how can I find the rotation outcome on a point like ...
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52 views

Proving that $S_{4}$ is not isomorphic to $D_{12}$

Question:Prove that $S_{4}$ is not isomorphic to $D_{12}$ This question seem trivial enough. But there is a subtle point that I feel isn't quite evident to me. It is trivial to see that both groups ...
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2answers
52 views

what does $\ltimes$ in the context of representation theory mean?

I am considering the following sentence wich is part of a theorem: '' Let $V$ be a finite dimensional unitary representation of $H=\mathbb{Z}^{2} \rtimes $ SL$_2(\mathbb{Z})$." I have no background ...
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3answers
52 views

Problem in solving a related to centre of a group.

Let $G$ be a group of order $8$ and $x$ be an element of $G$ then $x^2 \in Z(G)$,the centre of the group $G$. My work : If there exists an element of $G$ of order $8$ then $G$ is cyclic and hence ...
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Is every finite group a normal subgroup of a symmetric group?

By Cayley's theorem, we know that for any finite group $G$, there exists $N \in \mathbb{N}$ such that $G$ is isomorphic to a subgroup of $S_N$, the symmetric group on $N$ letters. Can we prove that ...
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3answers
80 views

An erroneous application of the Counting Theorem to a regular hexagon?

I'm trying to count the unique orbits of a regular hexagon such that each vertex is either Black or White and each edge is either Red, Gree, or Blue. The group I've chosen to act on the hexagon is the ...
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2answers
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Rebracketing Theorem

My questions regarding the below theorem Both questions are centred on Eq(2) and the paragraph preceding it. 1) How is it that Eq(2) contains $a_k$ but in that section of the proof the assumption is ...
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3answers
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What is one isomorphism?

Sorry if this is a silly question, but I had a Maths exam today and it asked me to show that 2 groups were isomorphic by "showing one isomorphism" between them. I simply showed the identity element ...
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Is the notion of Stabilizer of a subset A of a group G absurd?

Is the notion of Stabilizer of a subset,A of a group G is absurd? I don't know whether this makes sense or not,but for curiosity i want to know view of experts. UPDATE i'm dealing with ...
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29 views

Number of orbits with Burnside's lemma

We color a equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the ...
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1answer
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Find Sylow subgroups of $G=Aut(\mathbb Z_{276})$ that is the group of automorphism $\mathbb Z_{276} \to \mathbb Z_{276}$

It was in the test. I did: $Aut(\mathbb Z_{276}) =~ Aut(\mathbb Z_{23}) \times Aut(\mathbb Z_4) \times Aut(\mathbb Z_3) =~ U(\mathbb Z_{23}) \times U(\mathbb Z_3) \times U(\mathbb Z_4) =~ \mathbb Z_{...
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1answer
17 views

How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...
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Show that $U(8)$ is Isomorphic to $U(12)$.

Question: Show that $U(8)$ is Isomorphic to $U(12)$ The groups are: $U\left ( 8 \right )=\left \{ 1,3,5,7 \right \}$ $U\left ( 12 \right )=\left \{ 1,5,7,11 \right \}$ I think there is a bit of ...
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If $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$

Let $G$ a finite group and $G' \subseteq G$ the smallest normal subgroup of $G$ such that $G/G'$ is abelian. Prove that if $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$ My attempt: If $G$ is not ...
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1answer
23 views

Sylow subgroup of a symmetric group

consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutation.Let H be a 7-Syowl subgroup of$A_{20}$.Is that H must be cyclic? And is this statement correct ...
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1answer
11 views

Existence of a $G$-invariant metric on a principal bundle

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging ...
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Not getting how to prove reverse hypothesis.

This is a theorem from Dummit & Foote text- Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by $a \sim b$ iff $a=g.b$ for some $g \in G$ is an ...
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1answer
20 views

Converse of Lagrange Theorem for p-groups.

I need clarification on my understanding of Sylow Theorems. Can I say that a finite p-group will have a subgroup for each prime power? If the above is valid, can I say then that p-groups satisfy the ...
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If $H$ and $G/H$ are $p$-groups then $G$ is a $p$-group.

Please verify: If $H$ is a $p$-group, then $|H| = p^r$, for some integer $r$. If $G/H$ is a $p$-group, then $|G/H| = p^s$, for some integer $s$. But the cardinality of a quotient set is the index, ...
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1answer
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Condition in a theorem of Hall

There is a well-celebrated theorem of Hall, which characterizes solvable groups according to the existence of Hall-$\pi$ subgroups. In this theorem, I was wondering whether it can be stated in a ...
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Groups of order $25$

Please verify my solution that there are only two groups of order $25$ up to isomorphism. As $|G|$ is a prime squared, then $G$ is abelian. Since the Theorem of Finite Abelian Groups, $G$ is a direct ...
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Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
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Are there at least $3$ groups of order $16$ that has element of order $8$?

Are there at least $3$ groups of order $16$ that has element of order $8$? I know that probably the simplest way of doing this problem is looking at the element structure of the abelian groups of ...
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Clarification on a concept involving Hall subgroups

Suppose that $G$ is a finite group and $H\leq G$. If $Q \in$ Hall$_{\pi}(G)$ such that $Q \cap H \in$ Hall$_{\pi}(H)$. Is true that $Q \leq H$ when Hall$_\pi(H)$ $\subseteq$ Hall$_\pi(G)$
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Proof/Reasoning why the sgn function which counts inversions has the following property?

$\mathrm{sgn}(\pi\circ\sigma)=\mathrm{sgn}(\pi)\cdot \mathrm{sgn}(\sigma)$ I am familiar with how to count inversions and any insight for why this formula holds true would be very helpful.
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If C/A is abelian, then C/B is abelian.

I am trying to prove that if groups A $\lhd$ B $\lhd$ C, A $\lhd$ C, and C/A is abelian, then C/B and B/A are abelian. Clearly, B/A is abelian since it is a subgroup of the abelian group C/A. ...
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Seeing composition table

Let p > 2 be any prime number. The non-zero integers modulo p form a group of order p - 1 with respect to multiplication modulo p. $$ [a][b] = [ab] = ab (mod p) $$ denoted Gp. The identity is ...