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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interchanges/transpositions (how to read)

I have came across this before and just again now, in the same form of which I'm struggling to understand. Although I know it's link to parity, as a perm group pi: $$ \pi = \begin{pmatrix} 0 & 1 ...
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How to determine when this is a well-defined homomorphism between cyclic groups

I've been having trouble with this concept. I'm not very familiar with cyclic group structure. Question: Let $Z_{36} = \langle x \rangle$. For which integers $a$ does the map $\phi_a$ defined by $\...
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2answers
33 views

Find $Aut(Z_{6})$

Question: Find $Aut\left ( \mathbb{Z}_{6} \right )$ Note that $\mathbb{Z}_{6}=\left \{ 0,1,2,3,4,5 \right \}$ Observe: $\forall k \in \mathbb{Z}_{6},k^{6}mod6=e=0$ Recall: Suppose $\phi$ is an ...
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Why is this isomophism of $PGL(2,\mathbb{Z})$ with a Coxeter group injective?

Let $W$ be a Coxeter group with generators $s_1,s_2,s_3$, where $m(s_1,s_2)=3,m(s_1,s_3)=2$, and $m(s_2,s_3)=\infty$. I understand that there's a surjective morphism $\varphi\colon W\to PGL(2,\...
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1answer
78 views

A group is simple if an only if its homomorphic images are the trivial group and G itself (up to isomorphism)

I need to prove the following: Let $G$ be a group. Then it's simple if and only if there is only surjective homomorphism $G \to G'$ for $G' = \{ e \}$ or $G' \cong G$. Not sure how to approach ...
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21 views

Composition series in subgroups which have normal series of arbitrary length (no finite maximal length of a normal series)

Warning: here I mean by "normal series" what sometimes called as "subnormal series". That is, a series of subgroups $G_i$ of a group $G$ such that $G_{i+1}$ is normal in $G_i$: $...\subset G_{n-1} \...
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3answers
28 views

Example of a communtative ring with two operations where the identity elements are not distinct?

I was introduced to the definition of a field today, as a communtative ring with two operations, like $\Bbb{R} = \langle R, +, -, \cdot, ^{-1} \rangle$ and all the usual axioms; commutativity, ...
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1answer
38 views

Isomorphism between multiplicative group modulo n and that of its factors

I am not entirely sure if this is true, but if it is, I would be done with a very important proof. Let $a$, $b$ and $d$ be pairwise coprime. Prove that: $$|(\mathbb{Z}/ab\mathbb{Z})^*/<d>_{ab}| ...
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Writing some algebras by generators and relation [on hold]

I want to write the following algebras by generators and relations: $$\mathbb{C}; \mathbb{C}^{2}; M_{2}(\mathbb{C}); L^{\infty}( O(2)/C_{k}) \;\text{and}\; L^{\infty}( O(2)/D_{k})$$ where $C_{k}$ and $...
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How many groups of order $2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2\cdot 13^2$ exist?

The calculation of the number of groups of order $$2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2$$ (result $81883$) takes already two hours with GAP. So, the calculation of the number of groups of order ...
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2answers
53 views

Is a formula for $gnu(2pq^2)$ known, where $q=2p+1\ $?

Let $p$ be an odd prime such that $q:=2p+1$ is also prime. Denote $g(p):=gnu(2pq^2)$ = number of groups of order $2pq^2$ upto isomorphy. The following table shows the first few values ...
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1answer
28 views

Is $\operatorname{Stab}(\lambda)$ generated by the simple reflections it contains, for $\lambda\in A_0$?

For a finite Weyl group, the stabilizer of an element in the fundamental domain is generated by the simple reflections of the Weyl group that is contains. Does the same still hold for the closure of ...
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21 views

If a conjugacy class intersect with its centralizer what can be said about its elements?

Suppose that $G$ is a finite group and let $x\in G$. If $y\in x^{G}\cap C_{G}(x)$, what can be said about the relationship of $x$ and $y$, or anything about $x$?
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2answers
64 views

Computing $|\operatorname{Aut}(G)|$ of a given abelian group

I had compute $|\operatorname{Aut}(G)|$ of a given abelian group. Now using the fact $(|G_1|,|G_2|)=1$ problem boils down to compute $|\operatorname{Aut}(\prod_{i} \mathbb{Z}_{p^{a_i}})|$ for a prime $...
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1answer
33 views

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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26 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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1answer
45 views

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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1answer
30 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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22 views

(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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2answers
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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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31 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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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
38 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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13 views

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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24 views

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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1answer
26 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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2answers
56 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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60 views

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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0answers
47 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
65 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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2answers
258 views

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
81 views

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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4answers
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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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4answers
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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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1answer
72 views

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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2answers
83 views

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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2answers
29 views

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 ...
3
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2answers
75 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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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
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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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1answer
18 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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1answer
28 views

Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
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1answer
12 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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1answer
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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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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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1answer
21 views

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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4answers
51 views

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
21 views

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, ...
4
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1answer
39 views

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 ...