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

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8
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4answers
320 views

What are the generators of $(\mathbb{R},+)$?

The real numbers as a group under addition are an infinitely-generated group. I'm not sure what those generators are though (or if it even makes sense to ask the question). For example, could we say ...
3
votes
1answer
123 views

Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types.

Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types. So, the orbit of $\sigma \in S_n$ is the set $S_n \sigma = \{ \tau .\sigma : ...
1
vote
1answer
87 views

Determining whether the following is homomorphism

I am trying to solve a question, I almost have the solution and have some few points to ask. Here is the question: Let $\Bbb Z$ be the additive group of integers and let $\Bbb R^*$ be the ...
1
vote
1answer
134 views

Proof that every subgroup of a free group is free (without using algebraic topology)

The fact that every subgroup of a free group is free follows quite naturally from the theory of covering spaces applied to graphs. Is it also possible to proof this theorem without topology, by using ...
5
votes
2answers
229 views

Determining whether two groups are isomorphic

I am reading "A First Course in Algebra", and there, I am trying to solve the exercises, but there is something i don't understand. How do we understand whether two groups are isomorphic or not? For ...
3
votes
1answer
227 views

Hall's Theorem of Solvable Groups (Conjugacy)

The problem is asking for a proof of Hall's theorem for finite solvable groups, namely that there exist Hall $\pi$-subgroups of $G$ a finite solvable group for any set of primes $\pi$ and furthermore ...
1
vote
1answer
148 views

Sp(1) and SO(3) Locally Isomorphic - Trouble with part of proof

My apologies if this is fairly basic. I'm trying to understand the proof that $Sp(1)$ and $SO(3)$ are locally isomorphic but are not isomorphic but have run into some trouble. Here's what my ...
1
vote
1answer
50 views

Image of the composition of a kernel with a cokernel.

Let $ h:H\to G $ and $ k:K\to G $ be two normal monomorphisms and let $ f:H\ast K\to G $ theire coproduct. It is always true that $ h\text {coker} k $ and $ f\text {coker} k $ has the same image?
1
vote
2answers
64 views

If $N \triangleleft G$ and $\mathrm{Aut}(N)=\mathrm{Inn}(N)$, show that $G=NC_G(N)$

I have been working on this problem, without success... Let $N \triangleleft G$ and assume that every automorphism of $N$ is inner. Show that $G=NC_G(N)$. Since every automorphism of $N$ is inner, ...
4
votes
1answer
128 views

One-parameter group not a group? Why?

So one-parameter group $G$ is defined with a continuous group homomorphism $\phi: \mathbb{R} \rightarrow G$. But according to the texts I read, they say that $G$ must be distinguished from groups as ...
4
votes
4answers
218 views

Showing $(NM)/M \cong N/(N\cap M)$ for $N,M \triangleleft G$

This is problem 2.7 #6 from the second edition of Herstein's Topics in Algebra. If $N,M$ are normal subgroups of $G$, prove that $(NM)/M \cong N/(N\cap M)$. Any hints in the right direction?
6
votes
1answer
555 views

Group of order $8p$ is solvable, for any prime $p$

Consider the following question: Show that a group $G$ of order $8p$ is solvable, for any prime $p$. I am kind of stuck, but here are my first attempts: I chose the series of subroups ...
1
vote
1answer
106 views

A doubt from Herstein text

In Herstein's text the mappping $\psi:G\to\mathcal A(G):g\mapsto T_g$ is said to be a homomorphism where $\mathcal A(G)$ is the group of all automorphisms of $G$ and $T_g$ the inner automorphism ...
7
votes
1answer
281 views

Example of non-abelian partially ordered group

What is a simple example of a non-abelian partially ordered group?
0
votes
2answers
61 views

“Lifting the centralizer”

Let $G$ be a finite group, $T\le G$ and $N\unlhd G$ with $(|N|,|T|)=1$. Clearly $T$ acts by conjugation on $G$ and $N$ is a $T$-invariant subgroup; for this reason $T$ induces naturally an action on ...
1
vote
2answers
181 views

An alternative definition of a group?

Will the following definition of a group work as a basis for group theory: $\forall G,f,i,e (Group(G,f,i,e)\leftrightarrow f:G\times G\rightarrow G$ $\wedge i:G\rightarrow G$ $\wedge \forall ...
5
votes
2answers
220 views

Upper bound for the sum of the orders in a finite group

This is a cute question that I found in ML. I thought a little bit about it and couldn't complete it to a solution. Let $G$ be a non-cyclic finite group of order $n.$ Let $S(G)=\sum_{g \in G} ...
1
vote
2answers
174 views

$GL_n(\mathbb{R})$ is the union of two path connected subsets (Michael Artin's Algebra).

I have that $SL_n(\mathbb{R})$ is path connected and generated by elementary matrices of the form $I + a e_{i,j} (i \neq j)$, where $e_{i,j}$ is the matrix with $1$ at position $(i,j)$ and zero ...
4
votes
2answers
1k views

A group of order 30 has a normal 5-Sylow subgroup.

There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me. Lemma Let G be a group of order 30. Then the 5-Sylow subgroup of G is normal. ...
0
votes
2answers
85 views

Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$

I am trying to understand the proof of the following proposition: A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series. ...
4
votes
2answers
101 views

Frattini Subgroup of p-Groups

Letting $P$ be a $p$-group and $\Phi(P)$ be the Frattini subgroup of $P$ (the intersection of all maximal subgroups), the challenge is "Prove that $P/N$ is elementary abelian implies $\Phi(P)≤N$" ...
2
votes
2answers
266 views

Showing that if the center of a group is trivial, then the center of its automorphism group is trivial

I am trying to show that if for a group $G$ we have $Z(G)=1$, then $Z(\operatorname{Aut}(G))=1$. I have tried everything I can think of, but to no avail. The first part of the problem was to show ...
2
votes
2answers
222 views

A question about classifying a semidirect product of groups

I was trying to classify all groups of the form $\mathbb Z_{21} \rtimes_{\alpha} \mathbb Z_2$ and show that these groups are $\mathbb Z_{42}$, $D_{42}$, $D_6\times \mathbb Z_7$, and $\mathbb ...
3
votes
0answers
184 views

Which theorem did Poincaré prove?

Two related elementary facts in group theory are sometimes called Poincaré's theorems. If $H\lneq G$ and $[G:H]<\infty$, then there is $N\leq H$, $N\lhd G$ such that $[G:N]<\infty$. The ...
2
votes
3answers
55 views

Computing $\langle (13746) \rangle$ in $S_7$.

How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
4
votes
2answers
795 views

Proving that a group of order $112$ is not simple

So I'm proving that a group $G$ with order $112=2^4 \cdot 7$ is not simple. And I'm trying to do this in extreme detail :) So, assume simple and reach contradiction. I've reached the point where I ...
7
votes
1answer
739 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
1
vote
1answer
58 views

If $X(t)$ is a matrix valued path in $GL_n(\mathbb{R})$, then $X(t)^{-1}$ is also.

Suppose there is a continuous path $X:[0,1] \rightarrow GL_n(\mathbb{R})$ from $I$ to $A$. I need help showing that $X(t)^{-1}$ is continuous. This shows that there is a continuous path from $I$ to ...
0
votes
2answers
102 views

About some bijections

Let $H=ℤ^{r},\ r>0$, and $K=ℤ/nℤ,\ n>0$. Let $G$ be an abelian group such that $H,K$ are subgroups of $G$ with $G=H+K$ and $H\cap K=\{0\}$. Then there is an isomorphism $\phi:H×K→G$ defined by: ...
0
votes
1answer
39 views

Cyclic group of order 2012 and primitive roots

G is cyclic of order 2012 Is $g^{537}$ is also a generator of G? If $2012$ has no primitive roots does it says that no?
0
votes
2answers
174 views

Suppose $N\leq H\leq G$, and $N\lhd G$. Prove $H\lhd G\iff H/N\lhd G/N$

Let N be a normal subgroup of $G$ and let $H$ be a subgroup of $G$. If N is a subgroup of $H$, prove that $H/N$ is a normal subgroup of $G/N$ iff $H $ is a normal subgroup of $G$
5
votes
5answers
111 views

How to prove $x \in H$

How to prove that Let H be a normal subgroup of a finite group G. If $\gcd(|x|, |G/H|)$ = 1, show that $x \in H$.
4
votes
3answers
408 views

Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$

Let G be the multiplicative group $\mathbb{Z}_n^*$ for $n = 2^k$ and $k \ge 3$. Can we prove that no element has order bigger than $2^{k-2}$ ? My solution (not really a solution) : Since $n=2^k$, I ...
3
votes
1answer
105 views

Conjugation on subgroups of $A_4$ faithful?

Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$ I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
1
vote
1answer
148 views

How I can find the inverse of an isomorphism?

The motivation of this question can be found in Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not? My question is: How I can find the inverse of the ...
0
votes
3answers
124 views

Basic group theory proof: $a$ is fixed in $G$, $x$ is any element in $G$, show $x = x^{-1}$

$G$ is a multiplicative group with the identity element $e$, a fixed element $a \in G$ for which $axa = x^{-1}$, for all $x \in G$. Prove that $x = x^{-1}$. I have already proved that $a = a^{-1}$ ...
2
votes
0answers
54 views

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
0
votes
1answer
57 views

Finding a subgroup of Multiplicative group $\mathbb Z_{32}$

I am backing on some basic points about the multiplicative groups, like $\mathbb Z_{32}$ ,to review and I am really in a bad confusion to write the elements of a subgroup of it. For example, I want to ...
1
vote
0answers
42 views

Finite subgroups. [duplicate]

(Index of a subgroup $H$ in a group $G$ is the number of distinct left cosets of $H$ in $G$, it is denoted as $|G : H|$) How to solve that problem. Let $H$ and $K$ be subgroups of a finite group ...
-1
votes
1answer
47 views

What is the order of an element in $ℤ/2mℤ×ℤ/2ℤ$?

I know that the order of every $T∈ℤ/nℤ$ divides the size of the group $n$. My question is: What is the order of an element in $ℤ/2mℤ×ℤ/2ℤ$?
0
votes
1answer
49 views

Draw the lattice under a group isomorphism

Let $M$ and $N$ be normal subgroups of $G$ s.t. $G=MN$. Prove that $G/(M\cap N)\cong (G/M)\times (G/N)$. I have got the proof. But the question asks to draw the lattice. Is there any lattice? Since ...
1
vote
2answers
166 views

Group theory problem to be solved?

Let $G = S_n$, the symmetric group of order $n$, acting as permutations on the set $\{1,2,\dots,n\}$. Let $H = \{\sigma \in G \mid n \cdot \sigma = n\}$. (i) Prove that $H$ is isomorphic to ...
5
votes
2answers
171 views

Prove that $S_4$ has no subgroup isomorphic to $Q_8$.

The question is to prove that $S_4$ has no subgroup isomorphic to $Q_8$. Here is an answer. But what "then $H$ also contains all products of two 2-cycles" means in that answer? Thanks.
1
vote
2answers
912 views

Classify groups of order 6 by analyzing the following three cases:

(i) G contains an element of order 6. (ii) G contains an element of order 3 but none of order 6. (iii) All elements of G have order 1 or 2. I've got: (i) $G$ is the cyclic group of order 6, ...
2
votes
4answers
109 views

Prove $\left \langle (1 \;3), (1\; 2\;3\; 4) \right \rangle$ is a proper subgroup of $S_4$

The question is to prove $\left \langle (1 \;3), (1\; 2\;3\; 4) \right \rangle$ is a proper subgroup of $S_4$. What is the easiest way of proving? Thanks.
4
votes
1answer
86 views

Groups and Lagrange theory

There are two subgroups $H_1$, $H_2$ of $G$, if $H_1\neq H_2$ then $\gcd(|H_1|,|H_2|)=1$. Prove that the order of $G$ is a prime number and the group is cyclic. I know from Lagrange that the order ...
1
vote
3answers
105 views

Sylow subgroup of the normalizer of itself

So I was given a finite group $G$, with $P \leq G $, P is a p subgroup of $G$, and $P\in Syl_p(N_G(P))$. I want to show that P is a Sylow p subgp of $G$. So I attempted a contradiction, supposing ...
-1
votes
2answers
111 views

Cyclic Group of order $8$

Let $G=(a)$ be a cyclic group of order $8$ and let $H=(a^4)$ be its subgroup of order $2.$ Find the coset representation of $G$ by $H$.
2
votes
1answer
158 views

Show the extension is not cyclic

Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
2
votes
2answers
57 views

For any normal subgroup $(aN)^n=(a^n) N$ holds

Prove this theorem Let $G$ be a group and $N$ a normal subgroup of $G$. If $a \in G$ and $n \in Z$, then $(aN)^n = (a^n) N$. I know I should prove this theorem in 3 cases where $n = 0$, $n>0$, ...