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

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Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Question: Is $dim(V^H) \le ...
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45 views

Help proving basic group properties

This is what I know: (G,.) is a group $$a^0=e \\ a^n=a^{n-1}a\\a^{-n}=(a^n)^{-1}$$ I need to prove for n and m integers $$i)\ a^{m+n}=a^ma^n\\ii)\ (a^m)^n=a^{mn}$$ For i), my attemp was trying to ...
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1answer
28 views

Maximal subgroup and representations (dual part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Let $g \in G$, $K = H \cap ...
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1answer
101 views

Knuth-Bendix completion algorithm: word problem

Can someone explain me how to set up an algorithm to find the 12 normal forms of the group $A_4$ by making use of the Knuth-Bendix completion algorithm? So we have that $RRR=1, SSS=1$ and $RSRS=1$. ...
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1answer
32 views

Groups Question

Let a and b be elements of a group (G,*). Show that $(a*b)^2 = a^2 * b^2$ iff $a*b = b*a$ I'm trying to prove the iff statement from left to right first. $$\begin{align} (a*b)^2 &= a^2 *b^2 \\ ...
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0answers
30 views

Polya enumeration theorem, general form

In classic Polya enumeration theorem we have group $G$ acting on $X$ in $Y^X.$ Is there some similar statement, where we have also group $H$ acting on $Y?$ My opinion is that's true, but I can't ...
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1answer
31 views

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
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58 views

Intersection of $p$-subgroup normalizer

Let $Q \leq S$ with $S$ a Sylow $p$-subgroup of $G$. I am interested in conditions that guarantee $$R_Q = \bigcap\left\{ N_{S^g}(Q) : g \in N_G(Q) \right\}$$ is equal to $Q$. For instance $Q=S$ ...
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121 views

The path to understanding Frieze Groups

What is the "Path" for understanding what Frieze Groups really are? Generally in mathematics, there is a is a path or "building blocks" approach to learning something. For example if I know how to ...
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38 views

What is the center of the group $S_3$ x $Z/6Z$?

I just don't know what to do here so far. All I can reason with is that since the center must commute with every element in the set, the center must contain the identity. But since $S_3$ can be formed ...
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34 views

If $G$ is a group of order $n$ and $(n,\phi(n))=1 $ then $G$ is cyclic. [duplicate]

let $G$ be a group of order $n$. And $\phi(n)$ the Euler totient function. If $\phi(n)$ and $n$ are relatively prime then $G$ is cyclic.
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37 views

Group Order and Least Common Multiple

Let $G_1,G_2,...G_n$ be groups. Show that the order of an elements $(a_1,a_2,...a_n)$ $\in$ $G_1 \times G_2 \times ... \times G_n$ is lcm($o(a_1),...,o(a_n))$ I know I need to use the fact that the ...
3
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1answer
91 views

Centralizer of unipotent element of simple groups

Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank. We know that ...
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35 views

The relation between two isomorphic subgroup

More precisely, if there are two embedding maps, $\phi_1: H \to G$, $\phi_2: H \to G$, are $\phi_1(H)$ and $\phi_2(H)$ conjugated? Thanks.
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Non-abelian finite groups with exactly $n$ normal subgroups.

Let $\mathfrak{N}$ be the class of all non-abelian finite groups and define $\nu: \mathfrak{N} \rightarrow \mathbb{N}_{\gt 1}$ by $\nu(G)=|\{{1} \leq N \leq G: N$ normal in $G\}|$. Is the map $\nu$ ...
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1answer
78 views

Is $\mathrm{PSL} ( 2, \mathbb{Q} )$ a simple group?

I am a new poster but I don't think this question has been asked before. Pardon me if it is.
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39 views

projective geometry and projective space

Let $V$ is a vectorspace over field $F_q$, we denote the set of all subspaces of $V$ by $\mathcal{P}(V)$. I saw some referencess they mentioned $\mathcal{P}(V)$ as a projective space and some ...
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1answer
40 views

Induction on exponent rule for groups

Prove: $(a^k)^m = a^{km}$ for all $k, m$ integers. Attempt: I am looking at the case where $m < 0$ and $k >0$. I am doing induction on a positive integer $m$, but $-m$ will become a negative ...
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1answer
132 views

$\chi(g)$ group character $\Rightarrow$ $\chi(g^m)$ group character

Let $G$ be a group of order $n$ and and $\gcd(m,n)=1$. Let $\chi:G\rightarrow\mathbb{C}$ be a class function and define $\chi^m\!: g\mapsto\chi(g^m)$. How can one show that $\chi^m$ is a character iff ...
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2answers
35 views

Group associativity

Is this valid in a group? $a(bc)= (ac)b$ by regrouping. I want to do induction on $k$ for $a^k\,a^l= a^{(k+l)}$, but when I do the inductive step I will have to regroup it as asked in my question. ...
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34 views

A Question about Cayley Graph [duplicate]

Petersen graph is http://en.wikipedia.org/wiki/Petersen_graph. It is not Cayley graph. How to prove. Can someone give a general method to judge a graph is or not a Cayley graph?
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126 views

Nonisomorphic groups of order 12.

I'm trying to find 4 groups of order 12, none of which are isomorphic to each other. Should i be trying external direct products? So far i have $A_4, \mathbb Z_{12},\,$ and $\,\mathbb Z_6\times ...
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1answer
153 views

Abstract algebra, prove that $(a^m)^n$ =$ a^{mn}$

Let $a$ be an element of group $G$. For any integers $m,n \in \mathbb{Z}$ ($m,n$ can be positive and negative). Prove that $(a^{m})^{n}=a^{mn}$, then show that $(a^{-1})^{-1} = a$ by using what we ...
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1answer
33 views

Group of Integers and group of rationals. [duplicate]

Are the additive groups $(\mathbb Z, +)$ and $(\mathbb Q, +)$ isomorphic? I know the group of integers is cyclic because $1$ generates all elements in $\mathbb Z$ , so $\langle1\rangle = \mathbb Z$. ...
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29 views

If $A_1, …, A_k$ are finite subgroups of $A$, an abelian group whose orders are pairwise coprime, then show the sum $A_1+…+A_k$ is direct.

So what we want is for any $g \in G=A_1 + ... + A_k$ the sum $g=n_1+...+n_k, n_i \in A_i$ must be unique. First, let $|A_i| = a_i$ for each $i=1,...,k$. Then we know that there are integers $r_i$ so ...
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2answers
64 views

Amalgamated product of groups

This is just a problem solving question. Let $G$ be a finitely generated group such that $G=A*_CB$ where $|A:C|=|B:C|=2$ and $A,B$ are finite. Show that $G$ has a finite index subgroup isomorphic to ...
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1answer
47 views

Transitive group action restricted to normal subgroup

Let $G$ be a finite group, and let $\Omega$ be a transitive $G$-space. Assume 1 $\neq H \unlhd G$ and that |$\Omega$| = $p$ where $p$ is prime, and $G \leq Sym(\Omega)$. Deduce that then $H$ must act ...
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50 views

Is the group of invertible elements isomorphic to Z_4

Problem: Is the group $g(8) = \{[1], [3], [5], [7]\}$ isomorphic to $\mathbb{Z}_4$ or to the symmetry group of the rectangle? Attempt: I know that $g(8)$ is isomorphic to $\mathbb{Z}_4$ because I ...
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1answer
57 views

Abelian group $G$ and $G/Z(G)$

I am taking an abstract algebra course and I am having a hard time understanding some group theory. In the course notes, it is stated that: If $G$ is an abelian group, then it is trivial that ...
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1answer
60 views

Collapsing a Factor to the identity element - Fraleigh p. 14 Theorem 15.8

p. 146: We should acquire an intuitive feeling for this theorem in terms of $\color{red}{collapsing}$ one of the factors to the identity element. p. 147 15.8 Theorem: $\hat{H} = \{(h, e) ...
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26 views

Clarification/Help needed: for $H\lhd G$, a normal subgroup of index $(G:H)=n$, prove that $\forall g \in G, g^n \in H$.

I've been looking at this question, but I'm not sure where to go with it. I'm all sorts of confused here, and I'd be grateful if someone could explain the question and how to answer it. Let $G$ be ...
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Intuition of Picture - Collapse, Factor Group, Homomorphism, Normal Subgroup - Fraleigh p. 144 Figure 15.1

Let $N \unlhd G$. In the factor group $G/N$, the subgroup $N$ acts as identity element. Regard N as being collapsed to a single element, to the identity element. This collapsing of N together ...
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5answers
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Isomorphism from $\langle Z, +\rangle$ onto $\langle Z, \ast\rangle$?

I'm trying to do 3.16 in Fraleigh's algebra book. Here it is: The map $f: Z\to Z$ defined by$ f(n) = n + 1$ for $n$ in $Z$ is 1-1 and onto $Z$. Give the definition of a binary operation $\ast$ ...
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69 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
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49 views

Abstract Algebra - Permutations

I'm asked to show that $(1,2,3) \in S_3$ generates a subgroup which is normal. I know that I could show it explicitly but that would be tedious. I think it may have to do with the fact that $(1,2,3)$ ...
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26 views

Proof that $(M,\cup)$ is a monoid but not a group

$M$ is a set of finite subsets of $\mathbb{N}$. As an example could be $\{\{0\},\{1\},...\}$ I need to proof that $(M,\cup)$ forms a monoid but not a group. I am not quite sure how I can proof that. ...
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33 views

Class of finite groups aren't axiomatizable - no equational class

Let K be the class of all finite groups. Show that it is not a variety (closed under H, S, and P) and therefore not an equational class. It sounds pretty reasonable that K is closed under H and S. I ...
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154 views

Homomorphisms from $D_4$ to $S_3$.

Find all homomorphisms from $D_4$ to $S_3$. We have $D_4 = \{e,r,r^2,r^3,s,sr,sr^2,sr^3\}$ (where $r^4 = e = s^2$) and $S_3 = \{e,(12),(13),(23),(123),(132)\} = \langle (12) (13) \rangle$. Let ...
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64 views

Order of subgroup generated by two commuting nilpotent elements

Q: Suppose $G$ is a group, $a,b \in G$ such that $e,a$ and $b$ are all distinct, $a^2 = b^2 = e$, and $ab = ba$. Prove $\langle a,b\rangle$ is a subgroup of $G$ of order exactly $4$. My thoughts: I ...
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50 views

$H$ is a normal subgroup of order $2$.

Let $G$ be a group and let $H$ be a normal subgroup of order 2. Prove that every element of $H$ commutes with every element of $G$. Since $H$ is a normal subgroup of order 2, $H$ contains two ...
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Findout if $\min(x,y)$ and $\max(x,y)$ are semigroup, monoid or group

I need to find out whether $A \oplus B := \max(A,B)$ or $A \ominus B := \min(A,B)$ form a semigroup, monoid or group for $\oplus : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and $\ominus : ...
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67 views

Listing left and right cosets

List the left cosets $(gH)$ and right cosets $(Hg)$ for $H = \langle (123) \rangle$, where $H \le G$ and $G = S_3$. My work so far: $G = S_3 = \langle (12) (13) \rangle = \{ e, (12), (13), (23), ...
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1answer
39 views

Homomorphism from C* into itself having ker = R>0

So I feel like I must be missing an easy example here... I'm trying to find a homomorphism from the multiplicative group $\mathbb{C}-\{0\}$ into itself such that its kernel is the positive reals. ...
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16 views

Modular Group Arithmetic with Primes

I need help understanding what this means: Working in sets $\mathbf Z_N^* = \{a \in \{0,1,...,N-1\} : gcd(a,N) = 1\}$ If I have a prime $p$ then I claim there is a value $k$ such that $g^x = ...
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34 views

A product of minimal normal subgroups is the product of some subcollection

I would appreciate a hint on how to go about solving the following problem. Let $X$ be a finite set of minimal normal subgroups of $G$ and write $K:=\prod X$ (the product of all members of $X$). a) ...
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2answers
37 views

Embedding monomorphism between Symmetric Groups

Suppose that $m$ and $n$ are positive integers, and $m<n$. Define $I:S_m \rightarrow S_n$ as follows: Given $\alpha \in S_m$, we let $\hspace{150pt}I(\alpha)(k)=\alpha(k) ...
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42 views

Infinite order elements of $G/G^k$

Let $G$ be a group such that $G^k:=\{g^k\mid g\in G\}\triangleleft G$. There exist $G$ and $k\in\mathbb{N}$ such that $G/G^k$ has an element of infinite order? It's easy to show that if $G$ is ...
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1answer
172 views

Nilpotency of higher dimension Heisenberg groups

I'm trying to understand higher dimension Heisenberg groups, defined as: $$H^{2k+1} = \langle a_1,\ldots,a_k,b_1,\ldots,b_k,c \mid [a_i,b_i] = c, [a_i,c] = [b_i,c]=1, [a_i,a_j] =[b_i,b_j] = 1, i \neq ...
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2answers
47 views

Prove that $C_G(Z(G)) = G$.

$C_G$ is the centralizer and $Z(G)$ is the center of $G$ if that wasn't clear. The only thing I can get from this is that $C_G(Z(G)) = Z(G)$ because if $gx=xg$ then $gag^{-1} = agg^{-1} = a$. I know ...
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66 views

Divisible groups, exercise from Rotman's theory of groups

The following exercise is from Rotman, An Introduction to the theory of groups, 4th ed, p324. "The following conditions on a group G are equivalent: (i) G is divisible, (ii) Every nonzero quotient of ...