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

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14
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Schur's Lemma in Group Theory

The analogue of well celebrated Schur's Lemma in group theory will be "If $G$ is a finite simple group, and $\phi$ is a non-identity homomorphism from $G$ to $G$, then $\phi$ is an isomorphism". ...
3
votes
1answer
82 views

Showing that if $N \le G$ is finite minimal normal and every simple homomorphic image is abelian, then N is elementary abelian

This is part 2 of this question. Unsurprisingly, I'm having some difficulty with it. Hints are much appreciated. Here's the setup again: "Let $N$ be a finite minimal normal subgroup of a group ...
2
votes
1answer
63 views

About commutators and center o a certain group

Let $G$ be a group and $H, K\unlhd\ G$ (normal subgroups), with $K\leq H$ such that $\left[H, Aut(G)\right]\leq K$. Why does this imply that $H/K\leq Z(G/K)$? $Z(G)$ denotes the center of $G$.
5
votes
0answers
444 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
5
votes
5answers
703 views

If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$

Prove that if $G$ is abelian then the set $H$ of all elements of $G$ that are their own inverses is a subgroup of $G$. Naturally in an abelian group, $ab = ba$ for $a, b \in G$, however I'm not ...
3
votes
1answer
86 views

Action of $S_7$ on the set of $3$-subsets of $\Omega$

Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem: Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
3
votes
1answer
377 views

Subgroups of $\mathbb{Z}^k$ of finite index $n$

I want to describe all subgroups in $\mathbb{Z}^k$ of finite index $n$. I have solved it for the case $k=2$. In $\mathbb{Z}^2$, each subgroup of index $n$ corresponds to a matrix $\left( ...
3
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1answer
239 views

Irreducible representation of tensor product

Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$. What are the irreducible representaions of $\varphi_2 \otimes \varphi_3$?
2
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0answers
72 views

The intersection of all $gA$ containing $x$,where $g\in G$ , is a block.

Excuse me for the previous problem in my post,i have a problem with my computer the question is: Let $G$ transitive in $ X$, $x\in X$ and $\emptyset\neq A\subset X$. Then $$\bigcap_{x\in ...
7
votes
3answers
562 views

Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
2
votes
2answers
225 views

All of the dihedral groups are factor groups of the infinite dihedral group.

Show that $\operatorname{Aut}(\Bbb{Z}) \cong \{\pm 1 \}$ and write $\alpha : \mathbb Z_2 \rightarrow \operatorname{Aut}(\Bbb{Z})$ for the nontrivial homomorphism. The semidirect product $\Bbb{Z} ...
0
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2answers
120 views

General Linear Groups with Homomorphisms [closed]

Let $G=\mathrm{GL}_n(\mathbb R)$ and $H=\mathbb R^*$. Let $\phi : G=\mathrm{GL}_n(\mathbb R) \rightarrow \mathbb R^*$ be the map defined by $\phi(A)=\det(A)$. Show that $\phi$ is a group ...
4
votes
3answers
313 views

Show a certain group is contained in a Sylow p-group.

Statement: Let G be a group and p a prime that divides $|G|$. Prove that if $K\le G$ such that $|K|$ is a power of p, K is contained in at least one Sylow p-group. I just started studying Sylow ...
2
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1answer
111 views

Show that the SU(2) group is a Lie group

How can I prove that the SU(2) is a Lie group with the Pauli matrices as generators?
3
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1answer
175 views

Group extension of $\mathbb Z_4$ by $\mathbb Z_2$

Let $f : G →\mathbb Z_2$ be an extension of $\mathbb Z_4$ by $\mathbb Z_2$. Suppose that the induced action $α_f :\mathbb Z_2 →\mathbb Z^{\times}_4$ carries the generator of $\mathbb Z_2$ to $−1$. ...
2
votes
1answer
127 views

Function spaces and transitive group actions

Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first. Let $B$ be a topological space and $G$ a topological group ...
4
votes
1answer
51 views

Reference request for ordered groups

I've been reading Pete Clark's notes on commutative algebra, and I especially liked section 17 on valuation rings, and ordered groups in particular. I'm looking for more introductory material ...
0
votes
1answer
83 views

Let $G$ be transitive.Then $\beta\in \operatorname{fix}(G_\alpha)$ implies $G_\alpha = G_\beta$

i am new in this forum. My question is about group actions We have a transitive action of $G$ and $\beta$ a element in the fixed points of the stabilizer of another element $\alpha$. Then $\alpha$ ...
2
votes
0answers
77 views

Universal coefficient theorem maps

Let $G$ be a group and $A$ a trivial $G$-module. The Universal Coefficient Theorem yields a split exact sequence $$0\longrightarrow {\rm Ext}(H_{n-1}(G),A)\longrightarrow H^n(G,A)\longrightarrow {\rm ...
7
votes
0answers
143 views

Orders of elements and homomorphisms.

Corollary 4.6.8 There is a group $G$ of order $n^3$ given by $G= \{b^ic^ja^k \mid 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with ...
0
votes
1answer
60 views

Is this $\mathbb{Z}_2^n$?

What group is formed by binary strings of a fixed length, $n$, and the XOR operation (^)? For example, we have: For $n=1$: A^B = B^A = B A^A = B^B = A For ...
0
votes
1answer
126 views

the general linear groups - solvable

I'm doing some experimental mathematics and I'm in the situation where I need an answer to the question: Consider the general linear groups $\operatorname{GL}(2,10)$, $\operatorname{GL}(4,10)$ (the ...
8
votes
2answers
44 views

The index of $\xi_4^*$ in $\xi_4$

Just seeing if i'm right: With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative ...
2
votes
1answer
60 views

Proving that $X$ is a subgroup of $G$

If we're given that some $X\subset G $ such that $e\in X$, and $\forall g \in G$, cosets $gX$ partition $G$, is $X$ a subgroup of $G$? I'm not quite sure what it is I have to do. If $X$ wasn't a ...
1
vote
0answers
80 views

Relationship between curls, gradients, and divergences; and the Isomorphism Theorem

I am trying to develop a geometric intuition for the relationship between the curl, the gradient, and the divergence based on the Isomorphism Theorem, where the Isomorphism Theorem says that "If ...
1
vote
1answer
71 views

Groups of transformations

I tried to find literature and articles about groups of transformations, but mostly what I found is either about groups or about transformations. Can you suggest me literature where groups of ...
3
votes
1answer
98 views

Infinite products of a (finite) group

So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
0
votes
1answer
279 views

Is the the number of generators of a group the number of different generators that one finds if one counts over every generating set of the group?

Consider the additive group of integers as an example as mentioned at the bottom of the Wikipedia article. There are two generating sets that are mentioned; The set consisting of the number 1, {1}, ...
0
votes
2answers
130 views

Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$

Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$. Here is what I have here: Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
12
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3answers
597 views

What does an outer automorphism look like?

I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...
6
votes
1answer
181 views

Order of elements in a group.

Corollary 4.6.8. There is a group $G$ of order $n^3$ given by $G = \{b^ic^ja^k | 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with $c$, ...
2
votes
1answer
76 views

On the Cyclic Subgroups

I recently read in a book on group theory that, a (non-abelian) $2$-generator group all of whose proper subgroups are cyclic, has been constructed by Ol`sanski; these are infinite simple groups. (see ...
2
votes
1answer
52 views

Minimal Polynomials Annihilating an Abelian Torsion-Free Group

Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. ...
5
votes
0answers
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Why are the p-adic integers a linearly ordered group? [duplicate]

In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group. I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
2
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3answers
456 views

All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $ [duplicate]

All group homomorphism from $ \mathbb{Z} _m $ to $ \mathbb{Z}_n $ How could I find every group homomorphism?
4
votes
2answers
124 views

Finite abelian $2$-group

If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors). Any ...
2
votes
1answer
493 views

Commutative ring with unity Proof on the set of units?

the question is as follows (TRUE or FALSE.) If R is a commutative ring with unity, then the set of units in R forms a subring. (If true, give a short proof. If false, give a specic counter-example.) ...
2
votes
1answer
130 views

Prüfer groups are countable

I have read that for any prime number $p$ the Prüfer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
5
votes
3answers
169 views

Finding the subgroup of $(\mathbb Z_{56},+)$ which is isomorphic with $(\mathbb Z_{14},+)$ by GAP

I am sorting some easy questions for the students in Group Theory I. One of them is: Is $(\mathbb Z_{14},+)$ isomorphic to a subgroup of $(\mathbb Z_{35},+)$? What about $(\mathbb Z_{56},+)$? I ...
1
vote
1answer
105 views

Set of left cosets is a group

Let $N$ be a normal subgroup of $G$. Let $G/N = \{gN : g \in G \}$. Show that the $(G/N, \circ, 1_{G/N})$ is a group, where: $\circ : G/N \times G/N \rightarrow G/N$ is defined by $(gN \circ hN) = ...
2
votes
1answer
108 views

Zero divisors in crossed group rings

It is not difficult to see that a group ring $K[G]$ ($K$ a domain) has non-trivial zero-divisors whenever there exists a non-trivial torsion element $g\in G$. [In fact, in this case, $1-g$ is such a ...
2
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1answer
131 views

What is the reason for the name *left* coset?

Let $G$ be a group and let $H \leq G$ be a subgroup. It seems that it is now standard to call the cosets $$gH=\{gh \ | h \in H \}$$ the left cosets of $H$ in $G$. I have to admit to being slightly ...
2
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1answer
284 views

Understanding the internal direct product of a group.

I have come across the statement: $G$ is the direct product of its subgroups $N_1$ and $N_2$ if the following conditions hold: 1) $N_1,N_2$ are normal subgroups. 2) $N_1\cap N_2=\{e\}$ 3) They ...
3
votes
1answer
52 views

Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism.

Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism, where $s : K \rightarrow H \rtimes_{\alpha} K$ is given ...
0
votes
2answers
277 views

Rings | Homomorphisms | Units

Question Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$. Attempt ...
0
votes
2answers
44 views

how do I calculate the ismorphism group of a six-nodes-tree?

How do I calculate the ismorphism group of a connected six-nodes-tree? The tree has a node centred and the other 5 nodes are leaves of the graph. I already know the answer is 6, which is the quotient ...
0
votes
1answer
59 views

ordering of a group

An ordering of a group $G$ is a linear ordering $<$ on (the underlying set of) $G$ that satisfies, in addition, $a < b \implies ac < bc ∧ ca < cb$, for all $a,b, c \in G$. Show that a ...
4
votes
1answer
119 views

sylow basis of finite solvable groups

Let ‎‎$G‎‎$‎ be a‎ ‎finite ‎solvable non-$p$-group ‎and ‎‎$‎‎A$ ‎be a ‎‎‎‎maximal ‎subgroup ‎of ‎‎$G‎‎$‎. ‎ Therefore ‎$A‎‎$ ‎is ‎of ‎primary ‎index ‎‎$p^{n}‎ $‎‎, that is ‎$|G : A|=p^{n}‎$ ‎‎where ...
2
votes
2answers
70 views

question on subgroups of prime order

Let $G$ be a group and let $\,H,\, K\,$ be subgroups of $\,G,\,$ each of order $\,p,\,$ where $\,p\,$ is prime. Show that either $\,H\cap K =\{e\},\,$ or $\,H=K.\,$ Is the result true if ...
3
votes
2answers
92 views

On the eigenvalues of a linear transformation $\tau$ such that $\tau^3 = \mathrm{id}$

I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph. [..] we will start our analysis of an arbitrary representation $W$ of $S_3$ by looking ...