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.

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248 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 ...
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1answer
114 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 ...
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1answer
94 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 ...
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1answer
62 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 ...
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2answers
96 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 $N \setminus Z(G)\neq \langle e \rangle$.
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3answers
105 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
212 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
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1answer
273 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
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1answer
181 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
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1answer
109 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 ...
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0answers
63 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.
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2answers
89 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
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1answer
226 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 ...
4
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0answers
70 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)
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2answers
96 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 ...
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2answers
75 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
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1answer
159 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
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1answer
103 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 ...
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1answer
104 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
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1answer
58 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
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3answers
438 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
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1answer
56 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 ...
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2answers
88 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 ...
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1answer
75 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 ...
7
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2answers
308 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 ...
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2answers
89 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 ...
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1answer
575 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 ...
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2answers
972 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 ...
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3answers
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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
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1answer
95 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
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2answers
145 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 ...
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1answer
45 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{ ...
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5answers
429 views

Prove that the order of any element in the additive group of integers modulo n is a divisor of n.

I am not opposed to struggle but I have been on and off of this problem for three days and need to present the proof tomorrow. I am thinking that because I know for any element in the additive group ...
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2answers
118 views

Center is never a maximal proper subgroup [closed]

Prove that center is not a maximal proper subgroup of group $G$.
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1answer
87 views

$S_n$ and its subgroups

Show that $A_n$ is unique in $S_n$ with index $2$. I'm trying to use Quotient Group and Lagrange's Theorem to approach this problem but I'm still clueless. Can anyone tell me how to do this problem? ...
0
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2answers
81 views

Consider the matrices $A$ and $B$. Let $Q=\langle A,B\rangle$. Prove $|Q|=8$

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 $|Q|=8$ The ...
2
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0answers
256 views

Symmetries of a regular tetrahedron

Let $G$ be the group of symmetries of a regular tetrahedron $T$, including orientation-reversing symmetries. (a) Decompose the set of faces of $T$ into orbits, and describe the stabiliser of a face. ...
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1answer
49 views

Given a non-abelian group $G$ with $|G|=p^3$ and $p$ prime, how do I show that $|Z(G)|=p$?

Given a non-abelian group $G$ with $|G|=p^3$ and $p$ prime, how do I show that $|Z(G)|=p$? $Z(G)$ is as always center of $G$. It is easy to see that $|Z(G)|\in\{p, p^2\}$, but how do I exclude $p^2$. ...
2
votes
1answer
703 views

Representation Theory of the Dihedral Group $D_{2n}$

So I'm pretty new into Representation Theory having so far covered only a couple of example sheets. I'm thinking about the following question: Suppose we have the group $D_{2n}$ (for clarity this is ...
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0answers
70 views

Is this proof that $|S_n|=n!$ correct?

I've always heard that it's trivial that $|S_n|=n!$ where $S_n$ is the symmetric group of degree $n$. Now, my proof was the following: consider $I_n=\{1,\dots,n\}$ and consider ...
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3answers
173 views

Finite group of two generators

My question is simple : Any finite group of two generators is cyclic, semidirect sum, or direct sum ?
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4answers
371 views

If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?

Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$? Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using ...
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1answer
149 views

Number of orbits of a subgroup of the symmetric group $S_9$

Let $H=\langle (3\quad 4\quad 5),(1\quad 2\quad3)(7\quad8\quad9)\rangle \le S_9$ be a subgroup in $S_9$. Find number of orbits and their order. First I noticed $\mathrm{orb}(6)=\{6\}$. I also ...
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0answers
50 views

Extension of transvection

I need your kind help in following: Let $V$ be a finite dimensional vector space. Then $1 \neq \tau \in GL(V)$ is said to be a transvection if there is a hyperplane $W$ in $V$ such that $\tau|_W=I_W$ ...
3
votes
1answer
263 views

Surjective Homomorphism to $\mathbb{Z}$ -> pre-image has normal Subgroup of index n

Let $N$ be a normal subgroup of Group $G$ and $\phi: G/N \rightarrow \mathbb{Z}$ a surjective homomorphism. I have to prove, that for every positve integer $n\in \mathbb{N}$ exists a normal subgroup ...
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1answer
43 views

Groups and vectorspaces

Let $(G, +)$ be a commutative Group with the neutral Element $e$ and $\mathbb{Z}_2$ be the field of remainders mod 2. The multiplication is declared as $\star : \mathbb{Z}_2 \times G \rightarrow G : ...
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1answer
69 views

Cyclic group mapping

Let $G$ be a cyclic group of order $7,$ that is, $G$ consists of all $a^i$, where $a^7 = e.$ Why is the mapping $\phi:a^i\to a^{2i}$ an automorphism of $G$ of order $3$? I know the group $G$ is ...
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1answer
90 views

Proving a map is an automorphism

Let $G$ be a group; for $g\in G$ define $T_g:G\to G$ by $xT_g = g^{-1}xg$ for all $x\in G.$ Prove that $T_g$ is an automorphism of $G$. Since $T_g$ is onto then for $y\in G$ let $x = gxg^{-1}$ ...
4
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2answers
147 views

Braid Groups Mapped to Symmetric Groups

How can I construct five elements in terms of the Braid Generators $\sigma_1 \sigma_2$ that are in the kernel of the homomorphism from the braid group on three strands to the symmetric group on three ...
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2answers
288 views

$A_5$ has no subgroup of order 15 and 20

Show that $A_5$ has no subgroup of order 15 and 20. I have been thinking about this problem for so much time but I'm still clueless. Can anyone tell me how to do this problem? Thanks. I ...