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

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3
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3answers
107 views

Does $Aut(A)\cong Aut(B)\cong Aut(A\bigcap B)$ imply $A=B$?

The question is in the title: given two mathematical structures $A$ and $B$ that we conjecture to be equal, is it sufficient to prove that $Aut(A)\cong Aut(B)\cong Aut(A\bigcap B)$, where $Aut(X)$ is ...
1
vote
0answers
42 views

Group operation continuous in the interval topology

I'm trying to prove the following: We have a DLO without endpoints M, and a group operation on M, which is continuous in the interval topology. I want to prove: if $b<c$ then for every $a \in M$ ...
0
votes
2answers
46 views

find a generator for the group $G =\{ f(x) = x+n\mid n\in \Bbb Z \}$ with the group operation being composition.

Another question from 'A book of Abstract Algebra' by Pinter. For each $n\in \Bbb Z$ define $f_n = x+n$. Then $f_n\in S_{\Bbb R}$, the symmetric set on $\Bbb R$. The group operation being ...
3
votes
1answer
34 views

Let $G$ be a group, then let $f :G\to G$ via $f(x) = x^2$. Now, is $f$ injective and/or surjective?

Let $G$ be a group, then let $f :G\to G$ via $f(x) = x^2$. Now, is $f$ injective and/or surjective? To this end let $f(x)=f(y)$. Then $$x^2 =y^2 $$ If we take a look at the group $( \Bbb Q ...
10
votes
7answers
423 views

Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group

Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group. This question is from the book 'Of Abstract Algebra' by Pinter. Now $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ containt 8 elements. ...
5
votes
3answers
85 views

Show that $\mathbb{Z}_{10}$ is generated by 2 and 5.

In the book 'Of Abstract Algebra' by Pinter the following question is asked: Show that $\mathbb{Z}_{10}$ is generated by $2$ and $5.\,$ Here, $,\mathbb{Z}_{10}\,$ is defined as the group of ...
0
votes
0answers
44 views

the Bass-Serre tree is a tree

If i have a graph of groups on Y, and Y has only one edge and two vertices, how can I prove that the Bass-Serre tree is a tree using the reduced forn theorem for amalgamated free products?
2
votes
2answers
136 views

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$? I have calculated that there are $24$ elements of order $10$ I know that in a cyclic subgroup of order ...
3
votes
2answers
242 views

Any group of order 12 must contain a normal Sylow subgroup

This is part of a question from Hungerford's section on Sylow theorems, which is to show that any group with order 12, 28, 56, or 200 has a normal Sylow subgroup. I am just trying the case for $|G| = ...
2
votes
0answers
59 views

Results on Hall $\pi$-subgroups - proof verification

I would appreciate verification of the following proofs. The problems are all closely related and the proofs are similar, so I think it's only appropriate to post them together (also, they are part ...
7
votes
1answer
855 views

A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$

This is problem 3 from Hungerford's section about the Sylow theorems. I have already read hints saying to use induction and that $p$-groups always have non-trivial centres, but I'm still confused. ...
2
votes
2answers
95 views

prove of disprove :'For every $x\in G$ there exists some $y\in G$ such that $x=y^2$, where $G$ is a group."

I am working on a question in the book: A Book of Abstract Algebra by Pinter. The question asks to prove or disprove the following statement: For every $x\in G$ there exists some $y\in G$ such that ...
4
votes
1answer
86 views

If $G, H, K$ are divisible abelian groups and $G \oplus H \cong G \oplus K$ then $H \cong K$

This is an exercise in Hungerford. But can somebody explain why is the following not a counter-example? Let $G$ be the direct sum of $|\mathbb{R}|$ copies of $\mathbb{Q}$. Let $K$ be the direct sum ...
0
votes
0answers
27 views

Group $G$ is solvable iff $\exists s>0: G^{(s)} = \{e\},$ where $G^{(1)}: = [G,G]$ and $G^{(i+1)}:= [G^{(i)}, G^{(i)}] $

May I verify if my proof is correct? Thank you. For a group $G$ and $g_{i}\in G$, define $[g_{1}, g_{2}]= g^{-1}_{1}g^{-1}_{2}g_{1}g_{2}$ and $[G,G]=<g^{-1}_{i}g^{-1}_{j}g_{i}g_{j}| ...
2
votes
2answers
138 views

Showing There Is Only One Non-Trivial Normal Subgroup.

Show that $H$ $=$ {$(id), (12)(34), (13)(24), (14)(23)$} is the only non-trivial normal subgroup for $A_4$. Looking at this question, what I started with was: By Lagrange's theorem, there are only ...
1
vote
1answer
961 views

The product (or composition) of permutation groups in two-line notation?

This question has been asked before, I know: Product of Permutations However, his did not resolve my problem. Here's an example I've been looking at, which is to find the product of two ...
-1
votes
1answer
49 views

Isomorphism of pairs of Groups [closed]

For each pair in a list, decide with proof if the groups are isomorphic. (a) $C_{2}\times C_{6}$ (b) $C_{4}\times C_{3}$ (c) $C_{2}\times C_{2}\times C_{3}$.
3
votes
1answer
190 views

Number of groups of order $p^n$, where $p$ is prime

for $n=1$, it is cyclic. so, the number is $1$ for $n=2$, it is Abelian. so, the number is $2$ for $n\geq 3$, I don't know. Can you recommend a book or link which can be helpful for understanding ...
1
vote
1answer
82 views

The group of rigid motions of an icosahedron.

Prove that group of rigid motions of icosahedron is isomorphic to $A_{5}$. Can you help me to prove this? What I have done is shown that the order of the group of rigid motions of icosahedron is ...
1
vote
1answer
83 views

Generator of a subgroup of a cyclic group

Let $G$ be a cyclic group, and let $x \in G$ be its generator such that $|x| > 1$. Suppose $H$ is a nontrivial subgroup of $G$. Prove that if $m$ is the minimum positive integer such that $x^m ...
0
votes
1answer
45 views

Inner automorphisms of $S_3$

How do I prove that $S_3 \simeq \wp(S_3)$? So I must show that the group of inner automorphisms of $S_3$ is isomorphic to $S_3$. I haven't been given many examples on how to do these types of ...
0
votes
1answer
79 views

Prove that $N \setminus Z(G)\neq \langle e \rangle$.

Let $G$ be a group with $\operatorname{ord}(G) = p^n$, where $p$ is a prime number, and if $N \neq \langle e \rangle$ is a normal subgroup of $G$, prove that $N \setminus Z(G)\neq \langle e \rangle$.
0
votes
3answers
102 views

Examples of a map involving group actions

Okay, this is a trivial question but I need some non-trivial examples of a map involving group actions. What I mean: Let $G$ be a group acting on a set $A$. Let $G'$ be another group acting on ...
5
votes
1answer
148 views

Any continuous group homomorphism $\mathbb{R}\to \mathbb{R}^n$ is $C^\infty$

Show that any continuous homomorphism $\mathbb{R}\to \mathbb{R}^n$, with respect to the usual abelian group structure, is actually $C^\infty$. My attempt: Let $\varphi$ be such a map. $$\lim_{h\to ...
5
votes
1answer
172 views

Manipulating quotients and direct sums for abelian groups

I'm studying Homology in Hatcher's Algebraic Topology. I feel that there is a gap in my group theory knowledge that is making me struggle with this chapter. In particular, the book (and material ...
3
votes
1answer
148 views

Automorphism of $\mathbb{Z}\rtimes\mathbb{Z}$

I'm looking for a description of $\mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z})$. I've tried an unsuccessfully combinatorical approac, does anymore have some hints? Thank you.
4
votes
1answer
89 views

Groups with a R.E. set of defining relations

Reading around I found the following two assertion: 1) Every countable abelian group has a recursively enumerable set of defining relations. 2) Every countable locally finite group has a recursively ...
0
votes
0answers
61 views

$N_G(K) \cap H$ commutes elementwise with $K$

Let $G$ be a semidirect product of $H$ and $K$ such that $H \trianglelefteq G$. How can I prove that $N_G(K) \cap H$ commutes elementwise with $K$. Thanks in advance.
2
votes
2answers
78 views

In a cylic group of order $12$, we can find an element $g \in G$ such that $x^2 = g $ has no solutions.

$$ \textbf{PROBLEM} $$ If $G = \{ g^n : 0 \leq n \leq 11 \} $. Then we can find an element $a \in G$ such that the equation $x^2 = a$ has no solutions. $$ \textbf{ATTEMPT} $$ My claim is ...
2
votes
1answer
107 views

Show that all groups of order 48 are solvable

I find that there are either $1$ or $3$ Sylow-$2$ subgroups, and $1, 4$ or $16$ Sylow-$3$ subgroups. I need one of them to be $1$, so that it is normal, so I can mod out by it and have $2$ $p$-groups ...
0
votes
0answers
40 views

Subgroup of index greater than 1 implies existence of normal subgroup whose index is greater than 1 [duplicate]

I'm trying to answer the following question: Suppose that $G$ is a group containing a subgroup of finite index greater than $1$. Show that $G$ contains a normal subgroup of finite index greater than ...
4
votes
0answers
60 views

Proving a group is cyclic [duplicate]

Let $G$ be a group of order $pq$, where $p,q$ are primes, $p < q$ and $q≢1$ (mod $p$); how do we prove that $G$ is cyclic ? (I have no idea)
2
votes
2answers
57 views

Find an example of three groups $E < F < G$ where $E$ is normal in $F$ and $F$ is normal in $G$ but $E$ is not normal in $G$

Find an example of three groups $E < F < G$ where $E$ is normal in $F$ and $F$ is normal in $G$ but $E$ is not normal in $G$. I am looking to prove this case by giving an example, and I am ...
0
votes
1answer
47 views

(Commutative Banach algebra) Prove that $G(\mathcal A)$ be an open set in $\mathcal A$.

UPDATE I have a problem: Let $\mathcal A$ be a commutative Banach algebra. Denote $G(\mathcal A)$ is the set of all invertible elements in $\mathcal A$. Prove the following assertions: ...
2
votes
1answer
92 views

Find a group G such that its derived series is a subnormal series but not a normal series.

Find a group G such that its derived series is a subnormal series but not a normal series. I have tried several groups G and still can't find a group G with this property. Is anyone know such an ...
0
votes
1answer
79 views

Quotient groups of $p$-groups

Suppose I am trying to show that a group $G$ is solvable and I gotten to having $Z(G)$ be a p-group and $G/Z(G)$. Now if I can show that $G/Z(G)$ is also a $p$-group, then both are solvable implying ...
0
votes
1answer
80 views

Abelian group with cyclic subgroup and cyclic quotient is generated by two elements

I have a number of questions that I think are related. I'm studying Algebraic Topology by Hatcher. I have essentially the same question as here. When talking about homology groups, the book says that ...
3
votes
1answer
52 views

How to show that $S_4=A_3D_4$ and $S_4=A_3K$?

Ok, so I know how to do this by direct computation, but that seems like it will take a long time, and that is probably not how my professor wants this done. This is a practice question for the exam, ...
2
votes
3answers
242 views

Can an uncountable group be generated from a single element?

First question : can an uncountable group be cyclic? Ok so my though is if $G$ is generated by i then for $x\in G$ we have $x=i^n$ for integer n, so then it must be countable. Is there a way to ...
2
votes
1answer
53 views

Application of Main Homomorphism Theorem

This is related to this question. I just didn't want a prolonged discussion in the comments. Let $\phi: G \to G'$ be a homomorphism. Let $G$ be a finite group. Let $K \leq G$ be the kernel of ...
1
vote
2answers
78 views

Isomorphism theorem

Is it true the following result? Let $f:G \mapsto G^\prime$ be a surjective morphism of groups, and let $H$ be a normal subgroup in $G$. Then $f(H)$ is a normal subgroup in $G^\prime$ (this part is ...
0
votes
1answer
64 views

number of automorphisms for group in order 169

Let $G$ be a group with order 169. Prove number of automorphisms is at least 143. I thought that 169 is 13 squared so maybe G isomorphic to $ Z_{169} $ but I dont have any idea. How can I solve ...
5
votes
2answers
138 views

Finitely generated group which is not finitely presented

Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented? I was looking for a finitely presented group with a central ...
0
votes
2answers
64 views

For each pair in the list decide with proof if the groups are isomorphic

I have a question in my list of exercises and there is nothing in my lecture notes about it, and we havent done an example of anything similar. I missed a workshop due to illness so I fear I may have ...
1
vote
1answer
325 views

Factor group of a center of a abelian group is cyclic.

I am trying understand this proof http://www.proofwiki.org/wiki/Quotient_of_Group_by_Center_Cyclic_implies_Abelian but I am confused what its trying to prove. Wouldn't $G/Z(G)$ group have just one ...
0
votes
2answers
304 views

Prove that the centralizer subgroup is normal in the normalizer subgroup

To my dear friends with gratitude. I want to get help proving centralizer of a nonempty subset of a group is a normal subgroup in the normalizer of that set in the mentioned group.symbolically: $C_G ...
7
votes
3answers
587 views

If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$

If a group $G$ is isomorphic to $H$, prove that ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$. Can someone provide a step by step solution? Explaining along the way our strategy in proving this. ...
2
votes
1answer
94 views

Is there always $k \in \Bbb N$ such that $g^{k+1} \equiv g^k+1 \pmod p$, where $p$ is a prime number?

Let g be a generator of the group $\Bbb Z_p^*$. Show that there is a $k \in \Bbb N$ such that $g^{k+1} \equiv g^k+1 \pmod p$, where $p$ is a prime number. Excuse me please for bad interpretation of ...
2
votes
2answers
138 views

Prove that Q has an automorphism of order 3.

Let $$A=\begin{pmatrix}i & 0\\ 0 & -i\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}.$$ Let $Q=\langle A,B\rangle.$ Prove that Q has an ...
1
vote
1answer
43 views

$G$ abelian. If $G\cong \sum G_i$ then $mG \cong \sum mG_i$

Let $G$ be an abelian group and $m \in \mathbb{Z}$. If $G\cong \sum_{i \in I} G_i$, then $mG \cong \sum_{i \in I} mG_i$. $$\sum_{i \in I} G_i = \{ f:I\rightarrow \cup G_i \mid f(i) \in G_i \text{ ...