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.

learn more… | top users | synonyms (2)

0
votes
2answers
30 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 ...
5
votes
3answers
2k views

If $(ab)^3=a^3 b^3$, prove that the group $G$ is abelian. [duplicate]

If in a group $G$, $(ab)^3=a^3 b^3$ for all $a,b\in G$, amd the $3$ does not divide $o(G)$, prove that $G$ is abelian. I interpreted the fact that $3$ does not divide $o(G)$ as saying $(ab)^3\neq e$, ...
1
vote
1answer
67 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 ...
53
votes
10answers
2k views

Does $G\cong G/H$ imply that $H$ is trivial?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In ...
0
votes
0answers
13 views

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,\...
4
votes
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 ...
0
votes
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}| ...
2
votes
1answer
52 views

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 ...
2
votes
2answers
82 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 ...
0
votes
0answers
15 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} \...
2
votes
2answers
74 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 ...
2
votes
1answer
26 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 ...
-3
votes
1answer
71 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 ...
2
votes
2answers
1k views

Invertability of Singular 2x2 Matrix with all same real values.

Question: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...
0
votes
3answers
27 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, ...
3
votes
2answers
154 views

How can I calculate $gnu(17^3\times 2)=gnu(9826)$ with GAP?

I tried to calculate the number of groups of order $17^3\times 2=9826$ with GAP. Neither the NrSmallGroups-Command nor the ConstructAllGroups-Command work with GAP. The latter one because of the ...
0
votes
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 ...
3
votes
2answers
52 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 ...
0
votes
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 ...
4
votes
0answers
65 views

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 ...
-2
votes
0answers
36 views

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 $...
2
votes
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 $...
-1
votes
0answers
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 $x$?
0
votes
0answers
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 ...
2
votes
1answer
44 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 $\...
3
votes
2answers
238 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 ...
18
votes
1answer
479 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 : ...
2
votes
0answers
45 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 ...
4
votes
1answer
506 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 ...
0
votes
1answer
28 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 $...
1
vote
0answers
21 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]}\...
0
votes
2answers
30 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 $f\circ g \ne g\circ f$ ?
0
votes
1answer
64 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 $0$ ...
3
votes
0answers
72 views
+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}\...
2
votes
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 $f_A=xA^t$, ...
0
votes
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 \...
1
vote
0answers
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 ...
0
votes
0answers
11 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-...
5
votes
3answers
730 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. ...
3
votes
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 ...
-2
votes
4answers
92 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 ...
4
votes
3answers
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 ...
0
votes
0answers
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?
1
vote
0answers
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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
8
votes
2answers
256 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 ...
3
votes
3answers
82 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 ...
0
votes
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 ...