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

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2
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3answers
80 views

When does a ring map $R\to S$ produce a group epimorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be rings with $1$ (not necessarily commutative) and $f:R\to S$ a ring homomorphism preserving $1$. Let $\bar{f}$ be the ring map $M_n(R)\to M_n(S)$ given by $f$ acting on the matrix ...
3
votes
1answer
34 views

Can kernel of homomorphism tell you when a group action cannot be constructed?

I understand that for every action of group $G$ on a set $X$, there is a homomorphism: $$G\rightarrow S_X$$ It seems to me that this can be used to rule out many possible actions. For example, a group ...
0
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1answer
33 views

Basic requirements for inner semidirect products

I'm trying to consolidate my notes on inner semidirect products, and I would like to verify that the following statements are precisely correct. This is my attempt to clarify my understanding after ...
12
votes
3answers
705 views

Is there any intuitive understanding of normal subgroup?

As the define goes: A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation; that is, for each element $n$ in $N$ and each $g$ in $G$, the element $gng^{−1}$ ...
2
votes
3answers
44 views

Let $H$ be a subgroup of $G$ such that $\varphi(H) \subseteq H$ for every automorphism. Show that $H \triangleleft G$.

Let $H$ be a subgroup of $G$ such that $\varphi(H) \subseteq H$ for every automorphism. Show that $H \triangleleft G$. Let $Z(G)$ be the center of $G$. Show $\varphi(Z(G)) \subseteq Z(G)$ for all ...
0
votes
2answers
53 views

Let $\varphi:G \to K$ be an epimorphism. Let $J \triangleleft K$. Prove there exists a normal subgroup $H$ of $G$ such that $G/H \cong K/J$.

Let $\varphi:G \to K$ be an epimorphism. Let $J \triangleleft K$. Prove there exists a normal subgroup $H$ of $G$ such that $G/H \cong K/J$. By definition, an epimorphism has the following property: ...
4
votes
2answers
120 views

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2013}$?

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2013}$? Clearly, we can assume the Structure Theorem for finite abelian groups. Edited Later: All I ...
0
votes
1answer
29 views

Why $p|n$ and $r+sp=n^{p-1}\implies p|r$

Why $p|n$ and $r+sp=n^{p-1}\implies p|r$ The implication above appears in the proof of Cauchy's Group Theorem, but I don't see it why, does it require some knowledge of elementary number theory ? ...
-3
votes
1answer
31 views

What group are the group of symmetries for the shown figure

I have a question based on the post below:- What group are the group of symmetries of these figures isomorphic to - Fraleigh p. 85 Theorem 8.23, 24, 26 Fig (a) is isomorphic to Z2 as per the answer. ...
0
votes
1answer
112 views

Groups and Square Roots

Let $g \in G$ and $g$ has odd order $n$. I have already shown that there is a unique square root in $H=<g>$ by showing an isomorphism from H to H as $\phi(h)=h^2$. I now have to show that if $x$ ...
0
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1answer
36 views

Check: Prove that $T_b \circ T_c=T_{bc} \forall b,c \in G$

Let $G$ be a group, $H$ a subgroup of $G$, and $S=\{Ha|a\in G\}$. For $b\in G$, define $T_b:S \to S$ by $T_b(Ha)=Hab^{-1}$. Prove that $T_b \circ T_c=T_{bc} \forall b,c \in G$. Proof: ...
0
votes
2answers
90 views

How many subgroups or order 8 an abelian Group of order 72 can have

Let $G$ be an abelian group of order 72.How many subgroups of order 8 and 4 can have?? I have listed all possible abelian groups there are 6.Then i said that if im lookin for an abelian group of order ...
1
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0answers
65 views

The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$

Show that: The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$. I've tried to apply the first isomorphism theorem to this by defining the ...
1
vote
0answers
69 views

Does a ring map $f:R\to S$ induce a homomorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be commutative rings with $1$ and $f:R\to S$ a ring homomorphism. Does $f$ induce a group homomorphism $GL_n(R) \to GL_n(S)$? Progress I first consider the map $\bar{f}:M_n(R)\to ...
0
votes
2answers
34 views

If a subgroup H of a semidirect product intersects the kernel trivially, must H lie in a complement?

Let $G = N \rtimes K$ be a semidirect product (with $N$ normal in $G$). If $H < G$ satisfies $H\cap N = \{1\}$, must there exist a complement $M < G$ to $N$ such that $H < M$? By a ...
0
votes
2answers
38 views

Prove that group of $2$-nd roots of unity form a group.

I am not sure if I could explain the problem in the description but here it is: Let $C_n$ be group of all n-th roots of unity. Let $G=\{z\in C | z^2 \in C_n \}$.Check that $G$ is a group. Which group ...
0
votes
1answer
46 views

Action on $G$ by inner automorphism

I wonder something about an action of a group $A$ on a group $G$ by a automorphism; There are many nice result related with some restrictions such as when $(|A|,|G|)=1$ , $G$ is abelian or ...
3
votes
3answers
67 views

What group is $(\mathbb{Z}/24\mathbb{Z})^{*}$ isomorphic to

I want to determine which group $(\mathbb{Z}/24\mathbb{Z})^{*}$ is isomorphic to. $\mathbb{Z}/24\mathbb{Z}$ contains the 24 residue classes $z + 24\mathbb{Z}$ of the division mod 24. For brevity, I ...
2
votes
5answers
68 views

Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$

I am having a lot of trouble understanding the solution to this problem. $(gH)^n = g^nH \implies g^n \in H$ Why does $H^n$ just turn into the identity? I am very confused, any help is appreciated. ...
3
votes
0answers
20 views

Showing that $U$ is a normal subgroup of $N_G(U)$

I am given a group $G$, a subgroup $H \le G$ and the normalizer of $U$ in $G$, $N_g(U) = \{ g \in G: U^g = g^{-1}Ug = U\}$. I am asked to prove that $U$ is a normal subgroup of $N_G(U)$. Isn't this ...
1
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1answer
43 views

Show a bijection between two inverse images of a homomorphism

$ \Phi: G \rightarrow H $ is a group homomorphism. There are $ h,h' \in H $, so that $ \Phi^{-1}(\{h\})$ and $\Phi^{-1}(\{h'\})$ are not empty. Show a bijection between $ \Phi^{-1}(\{h\}) $ and $ ...
5
votes
1answer
80 views

Base for symmetric group

Given symmetric group $S_n$, is it possible to find $k=\lceil\log_2S_n\rceil=\lceil\log_2n!\rceil$ members $\{\alpha_i\}_{i=1}^{k}$ in $S_n$ such that every member of $S_n$ can be written as ...
0
votes
2answers
40 views

$|G|=p^3$, prove that $p$ divides |Z(G)|. [duplicate]

Suppose that a group $G$ has order $p^{3}$ where $p$ is prime. How would I prove that $p$ divides $|Z(G)|$?
1
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1answer
23 views

How do we compute |G| = |Ox| · |Gx|?

I was given a set X and a group G and was asked to find Gx and Xg. Then I was asked to find the G-equivalence class of X for each of the G-sets which is the orbit Ox but i'm having trouble verifying ...
1
vote
1answer
37 views

Group homomorphism between integers and real

Is there a non-trivial homomorphism $f:\mathbb{R}\rightarrow\mathbb{Z}$? I.e., there exists $ a\in\mathbb{R}$ such that $f(a)\neq0$
0
votes
1answer
119 views

Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can't find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the ...
2
votes
2answers
237 views

Are there any finite non-abelian group with one subgroup of each size ?

Let $G$ be a finite group with at most one subgroup of any size , then is it true that $G$ is cyclic ? I can prove that the answer is "yes" with the additional assumption of abelian ness on $G$ but ...
1
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1answer
62 views

Understanding Group Action

In general are we just supposed to make an educated guess about what the $G$-set is for a group action is if it's not specified? Here are two examples of what I mean. I am asked to find a fixed ...
1
vote
1answer
49 views

What are all the factor groups of $D_{4}$ up to isomorphism?

I know $D_{4}$ has $5$ subgroups and I've figured out the normal subgroups but I'm having trouble figuring out how the factor groups are isomorphic to $\mathbb{Z}_{4}$ and $\mathbb{Z}_{2} \times ...
7
votes
2answers
67 views

Embedding of finite groups into general linear group

It's clear that for any field $\mathbb{F}$ any finite group $G$ can be embedded into $GL_{n}(\mathbb{F})$ for some $n$. My question is about one modification of this result. Let's fix positive ...
5
votes
1answer
98 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
0
votes
1answer
68 views

In $S_n$, if $ε = α_1α_2 \cdots α_r$ where $α_i$ is a $2$-cycle, then $r$ is even.

In $S_n$, if $ε = α_1α_2 \cdots α_r$ where $α_i$ is a $2$-cycle, then $r$ is even. I don't know how to start. Note, $ε$ is the identity of the permutation group $S_n$.
1
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2answers
48 views

Group homomorphism, the uniqueness of k for g' = gk

Group homomorphism is $ \Phi: G \rightarrow H $ Show, that for all $ h \in H $ and all $ g,g' \in \Phi^{-1}(\{h\}) $ there exists a unique $k \in \ker(\Phi) $, so that $g'=gk$. $$ \forall h \in H, ...
2
votes
2answers
101 views

Every finite $p$-group is solvable

I know that in some version of Sylow's 1st theorem, it states that if $|G|=p^nm$ for some $n\geq 1$ and where $p$, a prime number, does not divide $m$, then every subgroup $H$ of $G$ of order ...
3
votes
1answer
27 views

For $H$ fixed point free show $|H| \leq n$

Suppose that $H \leq S_n$, and suppose that $H$ has the property that all non-identity elements of $H$ are fixed-point free. Show that $|H| \leq n$. I am trying to prove this by induction. For ...
1
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1answer
58 views

The natural isomorphism in the Pontryagin Duality

Pontryagin duality is the statement that there's a natural isomorphism between the identity functor on $\mathsf{LCA}$, the category of locally compact (Hausdorff) abelian groups with continuous ...
0
votes
1answer
43 views

Proving that in a Group the inverse of the inverse of an element is the element itself

Im trying to prove $(a^{-1})^{-1}=a^{-1}$.But a statement is confusing(Please see the highlighted portion in the image,i tried to type in the equation but its not working) . How can we say that the ...
1
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1answer
28 views

How does action by conjugation determine the product stucture of a semidirect product?

Consider $G=P\ltimes Q$ where $P\cap Q=\{e\}$ and $Q<N_G(P)$. Here $\ltimes$ is the inner semidirect product. Here I believe it is the case that the conjugation action of $P$ on $Q$ will determine ...
1
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1answer
24 views

Let $f$ be the mapping from $S_n$ to the additive group of $\mathbb{Z}_2$

$f$ is defined by $$f(\delta)= \begin{cases} [0],& \text{if $\delta$ is an even permutation}\\\\ [1], & \text{if $\delta$ is an odd permutation} \end{cases}$$ is $f$ an epiomorphism ...
4
votes
1answer
96 views

Let $G$ denote an arbitrary group. Prove: The center of any group $G$ is a normal subgroup of $G$

Let $G$ denote an arbitrary group. Prove: The center of any group $G$ is a normal subgroup of $G$ Let $G$ be a group and $C$ the center, i.e., for any $a \in C$ and any $x \in G$, $xa=ax$. So, ...
1
vote
1answer
93 views

find all distinct cyclic subgroups of A4

$A_4$ is the subgroup of $S_4$ consisting of all even permutations.I know that $A_4$ have 12 elements . $e = (1),a_1 = (1,2,4),a_2 = (1,4,2),a_3 = ...
2
votes
1answer
18 views

Proof that, if factor groups are equal, then the subgroups are equal

I'm reading a proof of the Correspondence Theorem and I'm seeing a step for which I'm not sure of the justification. So suppose that $G$ is a subgroup and $K< H_{1},H_{2} <G$, and ...
0
votes
1answer
45 views

Automorphism group of p group

Prove that the automorphism group of a group of order $p$ is cyclic. I have tried to solve this question for days but made no progress, can somebody help me with it?
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2answers
80 views

Classify $\mathbb{Z}\times\mathbb{Z}/\langle(2,2)\rangle$

I couldn't figure out how to solve this. I found another explanation for the same questions here and I didn't understand the hints. For the first coordinate I see that it has to be 0 or 1 being mod ...
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2answers
36 views

Consider $f: \mathbb Z_8 \rightarrow \mathbb Z_4$. List the cosets of the kernal of $f$

Consider $f: \mathbb Z_8 \rightarrow \mathbb Z_4$ given by $$\begin{pmatrix} 0&1&2&3&4&5&6&7 \\ 0&1&2&3&0&1&2&3 \end{pmatrix}$$ Verify the ...
3
votes
0answers
46 views

About motivation of Lang's Proof $S_n$ is not solvable for $n\geq 5$.

In Lang, he proves that $S_n$ is not solvable if $n\geq 5$ by using following observation. If $N\unlhd H\leq G$, H contains every 3-cycle, and if $H/N$ is abelian, then H contains every 3-cycle. Where ...
1
vote
1answer
60 views

Every $p$-subgroup is contained in one $p$-Sylow subgroup?

I am learning Sylow's theorems in my algebra course and I was reading questions posted before. One is the following: If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. ...
1
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1answer
40 views

Subgroups of Order $p^2$ in $\mathbb{Z}_p \oplus \mathbb{Z}_p$

Hello Mathematics Community. I am unsure about how to solve this problem involving the number of subgroups in an abelian group. How many subgroups of order $p^2$ does the abelian group $\mathbb{Z}_p ...
2
votes
2answers
84 views

Properties possessed by $H , G/H$ but not G

i) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are abelian but $G$ is not ? ii) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are cyclic ...
0
votes
0answers
32 views

List of functions $\chi_{s,a}(n)$ defined on a Group such that $\chi_{s,a}(n)\in{s,a}$ and depending on the parity

Question Let $(G,\cdot,e)$ be a non-commutative group and $s,a \in G$ .I'm looking for interesting functions $\chi_{s,a}:\Bbb N \rightarrow G$ witht this property $$\chi_{s,a}(n)= \begin{cases} s, ...