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

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73 views

Let G be a group with subgroups X,Y. Show that XY is a subgroup if either X or Y is a normal subgroup of G.

I've found and understood the proof for XY=YX but am unsure about this next bit. Show in particular that XY is a subgroup if either X or Y is a normal subgroup of G. I really don't know how to ...
1
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0answers
70 views

Terminology on group actions

Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. My knowledge of group theory is undergraduate-level stuff. I'm looking at the paper cited ...
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1answer
54 views

Determining a fixed point

Let $G$ be a group of order $14$, and let $S$ be a set of order $5$ on which $G$ operates. Prove that there is a fixed point, an element $s$ of $S$ that is left fixed by every element of $G$.
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1answer
31 views

Are these conditions enough to specify a unique number? (counting sylow p-subgroups)

Sylow's third theorem gives the two facts that the number of sylow p-subgroups $n_p$ of a group $G$, whose order we can write as $|G| = p^rm$ such that $p\not |\ m$ will satisfy both $n_p | m$ and ...
2
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4answers
490 views

Show that the integers have infinite index in the additive group of rational numbers.

Show that the integers have infinite index in the additive group of rational numbers. Anybody in a good enough mood to tell me how this is done?
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1answer
47 views

G-orbits for groups

What does it mean to find a decomposition of a set say S into G-orbits? Where G is our group. I am familiar with the usual orbit definition but I haven't come across G-orbit notation anywhere? Can ...
3
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1answer
60 views

Is it true that stabilizer in $O(n)$ of a rank $k$ matrix is isomorphic to $O(n-k)$?

Let $X\in M_{n, k}(\mathbb R)$ such that $\textrm{rank}(X)=k$ and,$$O(n)_X:=\{A\in O(n): AX=X\}.$$ Notice $O(n)_X$ is a subgroup of $O(n)$. Is it true that $O(n-k)\cong O(n)_X$? Here $O(n)=\{A\in ...
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3answers
194 views

Proving that two permutation groups are isomorphic

Here's the statement to prove: Let $n,m$ be two positive integers with $m≤n$. Prove that $S_m$ is isomorphic to a subgroup of $S_n$, where $S_n$ is the collection of all permutations of the set ...
3
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2answers
175 views

Do there exist functions such that $f(f(x)) = -x$? [duplicate]

I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective). ...
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1answer
48 views

Order of modular group

Prove $|(\mathbb{Z} / p^e \mathbb{Z} )^{\times}| = p^e - p^{e-1}$ I know it has something to do with the fact that we have $p^e$ elements and we're substracting $p^{e-1}$ multiples of $p$, but I'd ...
3
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1answer
104 views

Orbits of $\mathbb{Z}_n^{*}$ acting on a set $\mathbb{Z}_n$

Let $n\geq 2$ be an integer and consider the action $\Phi: \mathbb{Z}_n^{*}\times \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ defined as $$\Phi(\alpha)(x)=(\alpha x \textrm{ mod } n),$$ i. e. simply the ...
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1answer
136 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
2
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2answers
37 views

Subgroup $\{(1),(12)\}$ in $S_3$ is not kernel of any homomorphism

How do I prove the following statement Subgroup $\{(1),(12)\}$ in $S_3$ is not kernel of any homomorphism.
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2answers
979 views

Prove that a group of order 30 has at least three different normal subgroups

Prove that a group of order 30 has at least three different normal subgroups. Prove: 30=2$\cdot$3$\cdot$5 There are 2-Sylow, 3-Sylow and 5-Sylow subgroups. If $t_p$= number of $p$-Sylow-subgroups. ...
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0answers
73 views

The order of a group of automorphisms

I'm asked to find the order of the automorphism group of the group $\Bbb Z_2\times \Bbb Z_4$ and the group $\Bbb Z_4 \times\Bbb Z_4$ using presentations and by showing that $\Bbb Z_2\times \Bbb Z_4$ ...
1
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1answer
238 views

Homomorphic image of a Sylow p-subgroup is Sylow p-subgroup.

Let $G$ be finite group and $\phi$ : $G \to H$ be a group epimorphism. If $P$ is a Sylow p-subgroup of $G$ then $\phi(P)$ is a Sylow p-subgroup of $H$. It looks very simple but I'm stuck here. Let ...
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1answer
57 views

free object isomorphisms

In group category if $F_1$ be a free object on $X_1$ and $F_2$ is free object on $X_2$ and $F_1$ isomorphic to $F_2$ prove that |$X_1$|=|$X_2$| whats the relationship between isomorphisms of free ...
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1answer
175 views

There are only two types of groups of order $6.$

There are only two types of groups of order $6.$ Could anyone advise on how to prove a/m claim? Here is my attempt but I'm stuck: If $\exists g\in G$ such that $o(g) =6,$ then $G = \left ...
2
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1answer
115 views

Does a homomorphic image of even permutations consist of even permutations?

If $f:S_n \to S_n$ is a homomorphism, prove $f(A_n) \subseteq A_n$. If every image of a transposition is even, then there is nothing to prove, but it is not sure.. How can I prove the problem?
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2answers
52 views

Why $M_n$ $\not\cong$ $O_n\times T_n$?

I would like to know why $M_n$ $\not\cong$ $O_n\times T_n$, where $M_n$ is the group of isometries of $\mathbb R^n$, $O_n$ is the group of orthogonal matrices, and $T_n$ is the group of translations ...
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2answers
85 views

G acts on L. H is a minimal normal subgroup. H abelian, and acts transitively. Prove G acts primitively on L.

$G$ acts on $L$. $H$ is a minimal normal subgroup (As in, H contains no normal subgroups). $H$ is abelian, and acts transitively. Prove $G$ acts primitively on $L$. I'm hoping to get some advice on ...
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1answer
107 views

Power of a generator generates the group iff this power is coprime to the group order.

Let $\langle g\rangle$ be a cyclic group of order $n$. Suppose $1\leq q \leq n-1$, I want to show that $g^q$ generates $\langle g\rangle$ if and only if $\gcd(n,q)=1$. Suppose $g^q$ generates ...
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0answers
46 views

Homomorphism on the group of isometries

Prove that the map $f: M \rightarrow \{1,r\}$ defined by $t_a \rho_{\theta} \mapsto 1$, $t_a \rho_{\theta}r \mapsto r$ is a homomorphism. M denotes the set of isometries of the plane; r the reflexion ...
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1answer
68 views

List all subgroup elements of $\mathbb{Z}$ that contains $8$ and $12$

Problem : List all subgroup elements of $\mathbb{Z(*,+)}$ that contains $8$ and $12$. My solution is : $\mathbb{Z}_{13}$ since it contains $8$ and $12$, and then of course $\mathbb{Z}_{13 + k}$ where ...
4
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2answers
579 views

Questions on Möbius function

I was reading a paper, related to group theory. I came across two doubts (may be simple, but puzzling me): Let $G$ be a finite group and consider the lattice of subgroups of $G$. On this lattice, ...
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1answer
82 views

Can one prove $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ if and only if $gH = Hg$ for all $g \in G$?

Let $G$ be a group and $H$ a subgroup of $G$. I have proven in an exercise that $gH = Hg$ for all $g \in G$ implies $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ where $(g_1H)(g_2H) = (xy | x\in g_1H, ...
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1answer
63 views

Infinite groups of order $2$ [duplicate]

Find an infinite group in which every element $g$ not equal to $e$ has order $2$. In this case $e$ is the identity element. I have tried so many times but i can't really seem to understand what ...
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1answer
67 views

How to show $GL(n, \mathbb R)/GL(n-k, \mathbb R)\cong S_k(V)$..

let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can anyone help me showing the groups $GL(n, \mathbb R)/GL(n-k, \mathbb R)$ and $$S_k(V)=\{A:\mathbb R^k\rightarrow V: A\ \textrm{is linear ...
6
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3answers
101 views

Group element not taken to its inverse by any automorphism

What is an example of a group G with an element g such that no automorphism of G takes g to its inverse?
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1answer
65 views

Isomorphsim: Prove that $\varphi(C(a)) = C(\varphi(a))$

Let $\varphi:G\to\overline G$ be an isomorphism from a group $G$ to a group $\overline G$ and let $a$ belong to $G$. Prove that $\varphi(C(a)) = C(\varphi(a))$, where $C$ denotes centralizer.
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1answer
52 views

Suppose G is a non-abelian group and a∈G. Prove that if C(a)=<a> then a∉Z(G).

Suppose $G$ is a non-abelian group and $a \in G$. Prove that if $C(a) =<a>$ then $a \notin Z(G)$. We know that $C(a)$ is the centralizer of $a$ in $G$ and $Z(G)$ is the center of the group $G$. ...
3
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1answer
88 views

Extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$

Given that $H^2(\mathbb{Z}_n,\mathbb{Z})=\mathbb{Z}_n$, it follows that up to equivalence there should be $n$ extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$, one for each cohomology class. I'd like to ...
0
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2answers
123 views

Find the subgroup $H$ of $A_4$ generated by $(123)$

I need to find the subgroup $H \leq A_4$ generated by $(123).$ I know that this subgroup will have order 1,2,3,4,6, or 12.
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1answer
325 views

Example of a Non-Abelian Infinite Group

I was hunting an example of a non-trivial infinite group in which 1) All non-trivial normal subgroup are non-abelian. 2) There exists a nontrivial subnormal abelian subgroup. Is there any hope to ...
2
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1answer
79 views

If $G$ is a finite group, $H⊲G$, $G/H$ is finite $p$-group and $H$ cyclic, show that

If $G$ is a finite group, $H⊲G$, $G/H$ is finite $p$-group and $H$ cyclic, show that $H\cap[G,G]\subseteq Z([G,G])$ and $[[G,G],[G,G]]$ is $p$ -group
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2answers
76 views

Torsion-freeness of the group $\langle a, b \mid a b^m = ba^n\rangle$

For integers $m$ and $n$ let $K(m,n)$ be the group $\langle a, b \mid a b^m = ba^n\rangle$. Is there a special name for this group? Is there a complete characterization of those pairs $(m,n)$ for ...
4
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1answer
230 views

The conditions $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ make the group $G$ abelian?

A group $G$ satisfies $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$. Prove that the group is abelian. Also please tell me whether there is any standard approach in proving commutativity of groups like this ...
4
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3answers
121 views

Let $R$ be the set of all integers with alternative ring operations defined below. Show that $\Bbb Z$ is isomorphic to $R$.

For any integers $a,b$, define $a\oplus b=a + b - 1$ and $a\odot b=a + b - ab.$ Let $R$ be the ring of integers with these alternative operations. Show that $\Bbb Z$ is isomorphic to $R$. What I ...
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2answers
195 views

Sylow's Theorem Application. Prove $G$ is Abelian.

Assume that $|G|=5^27^2$. Determine the possibilities for $n_5, n_7$ and determine what can be concluded in each case about the $5$-Sylow subgroups and the $7$-Sylow subgroups and prove that $G$ is ...
6
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1answer
110 views

What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?

Evidently, $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ has order $4$, but I think it's infinite. The four cosets are listed as $(0,0) + \langle (2,2) \rangle$, $(0,1) + \langle (2,2) ...
4
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0answers
89 views

Embedding $\mathbb{G}_a$ into $GL_2$

Let $k$ be an algebraically closed field of characteristic $p$. I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is ...
5
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1answer
178 views

Example of a Non-Abelian Group

I was hunting an example of a non-trivial finite group in which 1) All non-trivial normal subgroup are non-abelian. 2) There exists a nontrivial subnormal abelian subgroup. Is there any hope to ...
7
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1answer
137 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
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2answers
601 views

coset and langrange's theorem problem

Prove that if $G$ is a finite group, the index of $Z(G)$ cannot be prime. First of all, I'm not sure what does $Z(G)$ mean. Can somebody tell me what does this symbol mean and how to solve this ...
3
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0answers
75 views

Showing $|K : H \cap K| \le |G:H|$ where $H,K \le G$

I'd appreciate input on the validity or lack thereof of my attempted proof of the following: Let $H,K \le G$ be subgroups, where $|G:H| < \infty$. a) Show that $|K:H \cap K| \le |G:H|$. b) Show ...
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1answer
150 views

Understanding what an action is?

This is a very simple question, and I am quite embarrassed to ask it! I'm trying to understand what an action is in general, and perhaps the best place to start is to try and outline my current ...
5
votes
3answers
230 views

Showing that the Class of Cyclic Groups Aren't Axiomatizable

The class of finite cyclic groups are not axiomatizable, for if we supposed they were by some set of sentences $\Sigma$, then there would exist a model for $\Sigma$ of at least order $n$ for all $n ...
3
votes
5answers
141 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
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2answers
90 views

50/5 = 10 via Quotient Groups

In this article the example of $\tfrac{50}{5} = 10$ example motivating quotient groups. I can see the idea, but I can't see any of the details really so I went & asked some friends, & some ...
3
votes
1answer
273 views

Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...