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

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Describe the cosets

$G$ is a cyclic group. $H$ is a subgroup of $G$. $|G|=12$, $|H|=3$. Why the sets of left and right cosets is a $H$, $xH$, $x^2H$, $x^3H$?
3
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2answers
66 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
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2answers
31 views

Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups

I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| > 1$. There has been a similar question on MSE here: How to show ...
2
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2answers
387 views

Is that Ring a field?

Given a commutative Ring $R$ of ordered pairs $(x,y)$ of reals $x,y$ with addition and multiplication defined in the following way. $$(x,y) + (u,v) = (x+u,y+v)$$ $$(x,y).(u,v) = (xu-yv,xv + yu)$$ I ...
4
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0answers
17 views

Definitions of Platonic and Archimedean Solids using Symmetry Groups?

A Platonic Solid is defined to be a convex polyhedron where all the faces are congruent and regular, and the same number of faces meet at each vertex. An Archimedean Solid drops the requirement that ...
2
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0answers
41 views

Why is the center of $SU(n) ≅ \Bbb Z_n$?

How can I show that the center of $SU(n)$ is isomorphic to $\Bbb Z_n$?
2
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2answers
32 views

If $G$ has no proper subgroup, then $G$ is cyclic of prime order

This is something I'm supposed to be able to prove for an upcoming test, but I can't find anything to help me prove this in my notes or the chapter, which is on cosets and Lagrange's theorem. If all I ...
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1answer
21 views

Showing that $|Z(G)| = p$ and $G/Z(G) \simeq \mathbb{Z}_p \times \mathbb{Z}_p$ for finite p-groups with order $|G|=p^3$

I have a finite, non-abelian $p$-group $G$ with $|G|=p^3$. I want to show that $|Z(G)| = p$ and $G/Z(G) \simeq \mathbb{Z}_p \times \mathbb{Z}_p$, where $Z(G)$ is the center of $G$. From the ...
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0answers
17 views

Rubik's Slide Proof's and Symmetries in a Rubik's Slide

$\quad$In the February edition of The Mathematical Association of America Monthly there is a article called "$\mathit{Rubik's\ on\ the\ Torus}$". Where they are dealing with solving problems involving ...
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3answers
44 views

If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$?

If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$? I know the proof that as rings, $\mathbb Z_{ab}$ is congruent ...
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0answers
39 views

Prove that $S_n$ is a group with respect to composition

Prove that $S_n$ is a group with respect to composition $fn = f\text{ follow }n$. I know that to prove something is a group I have to show identity, inverse, close, and associativity. But I do ...
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1answer
35 views

Suppose $G$ is a group, with a subgroup $K$ and a normal subgroup $H$.

Let $h, h_1 \in H$ and $k, k_1 \in K$. If $hk = h_1k_1$, show that $h_1=hb$ and $k_1=b^{-1}k$, for some $b \in H \cap K$. I noticed that $H \cap K$ is a subgroup of $K$, and $HK$ is a subgroup of ...
1
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0answers
26 views

$\{m\in\mathbb Z_{2^n}:m\equiv 1 \mod 4\}$ is cyclic

How can I prove that the multiplicative group $G:=\{m\in\mathbb Z_{2^n}:m\equiv 1 \mod 4\}$ is cyclic? I tried claiming that $G=\langle5\rangle$ but failed.
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1answer
20 views

Testing Normality in a Group

Let H be a normal subgroup of G. Assume that $ab \in H$. Show $ba\in H$. Consider the conjugate of $ab\in H$ with the element $b$. My question is what is the conjugate of a $ab$ with $b$? Also the ...
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1answer
14 views

images of subgroups of $G = \mathbb{Z}_4 \times \mathbb{Z}_4$ in $G/G[2]$

I am asked which subgroups of $G = \mathbb{Z}_4 \times \mathbb{Z}_4$ have the same image in $G/G[2]$, where $G[2] = \{ g \in G: \operatorname{ord}(g) \,|\, 2\}$). I have determined all subgroups and ...
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1answer
20 views

The converse of Factor Group criterion

If N is a normal subgroup of G then G/N is a group. Is the converse true? I mean If G/N is a group then N is Normal.
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2answers
68 views

Cyclic finite groups and their direct product

Let $C$ be a cyclic group with three elements. prove that $C \times C \times C$ can not be generated by two elements. I was thinking that showing that $C ^{3}$ is isomorphic to ${\mathbb{Z}_{3}} ^{3}$ ...
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0answers
42 views

Why is there no normal, dense, totally disconnected subgroup of $SO(n)$?

There are two exercises in Stillwell's Naive Lie Theory that I'm having trouble doing: 3.8.4: Show that the subgroup $H = \{ \cos 2\pi r + i\sin 2\pi r : r\text{ rational} \}$ of the circle $SO(2)$ ...
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2answers
15 views

Does having a subnormal series with abelian quotient imply being abelian?

Let $G$ be a group and suppose I have a normal series $$e = U_0 \subseteq U_1 \subseteq ... \subseteq U_n = G$$ with $U_{i-1}$ normal in $U_i$ and with each quotient $U_i/U_{i-1}$ abelian. Does ...
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0answers
30 views

Equivalence relation of sub-groups

Let $G$ a group and $H_{1},H_{2}<G$. $H_{1}\sim H_{2}$ iff $\left[H_{1}:H_{1}\cap H_{2}\right]<\infty,\left[H_{2}:H_{1}\cap H_{2}\right]<\infty$. I need your help proving the transitivity of ...
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2answers
25 views

Do isomorphisms preserve simplicity?

This is a very simple* question that I surprisingly can't find the answer to, and I am too stupid to come up with a counterexample or a proof. So does simplicity of one group and an isomorphism ...
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0answers
32 views

What are some of the effective methods and theorems to prove that a group or subgroup is abelian? [on hold]

What are some of the effective methods and theorems to prove that a group or subgroup is abelian? Can someone give me a list of them based on your experience. Thanks. Now I have always been trying to ...
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3answers
61 views

Trying to calculate the quotient group $\mathbb{Z}\times\mathbb{Z}/\langle (1,1),(1,-1)\rangle$ [on hold]

Let $G$ be the group $\mathbb{Z} \times \mathbb{Z}$ and $H$ be the subgroup of $G$ generated by $(1,1)$ and $(-1,1)$. I am trying to calculate the quotient group $G/H$.
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1answer
33 views

Isomorphism between $\operatorname{O}(A)$ and $\operatorname{SO}(A\times \{0\})$ for $A \subset \mathbb{R}^2$

I was given this exercise and to be honest I can't wrap my head around this one at all. Maybe some of you can shed some light on the problem at hand. I don't want a full solution, but some hints would ...
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2answers
97 views

How is number of conjugacy class related to the order of a group?

Let $c(G)$ denote the number of conjugacy classes of a group $G$. I have to show that $$\lim_{n \to \infty} \inf _{|G|=n}c(G)=\infty.$$ That is, I have to show that $\exists $ a function ...
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2answers
42 views

Prove that if there is a surjective homomorphism from $S_n$ onto $C_r$, then $r$ must be $1$ or $2$.

Prove that if there is a surjective homomorphism from $S_n$ onto $C_r$, then $r$ must be $1$ or $2$. I searched this on google, there is a proof using commutator subgroup, unfortunately, i don't know ...
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1answer
45 views

A question concerning 4-cycles in $S_4$

Is it true that for all $g\in S_4$ and $f \in S_4$ a 4-cycle, then $g^{-1}fg=h$ implies $h$ is also a 4 cycle. I did a few examples, and it seems to be true, but I don't know how to prove it. Also, I ...
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1answer
45 views

If $p_1, p_2,…,p_s$ are distinct primes, show that an abelian group of order $p_1p_2\cdots p_s$ must be cyclic

Can anyone give me some clues on this? Specific steps needed! Thank you!
2
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1answer
59 views

Prove either $G=ST$ or |$G|\geq|S|+|T|$

Let G be a finite group, and let S and T be (not necessarily distinct) nonempty subsets. prove that either $G=ST$ or |$G|\geq|S|+|T|$ That's my thougt, I am thinking suppose $G$ does not equal to ...
0
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1answer
14 views

Two quotient morphisms and universal property

I am reading some notes on group theory and I am having some doubts related to the following: Let $S \lhd G$ and let $\rho:G \to Q, \space \rho': G \to Q'$ be two quotients of $G$ by $S$. Then, by ...
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41 views
+50

Is there a specific name for this set of square-rooted primes?

Consider the set of all the primes numbers (± square rooted) and all the irrational numbers that can be formed under their addition (only the addition of finitely many elements is allowed, i.e. no ...
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0answers
44 views

Finding the normal subgroups in a semidirect product

Let the group $A=\cdots\times\mathbb{Z}_{-1}\times\mathbb{Z}_{0}\times\mathbb{Z}_{1}\times\cdots$ with $\mathbb{Z}_{i}=\left\langle a_{i}\right\rangle $ and $\alpha:a_{i}\rightarrow a_{i+1}$ an ...
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2answers
94 views

If the commutator of a finite group has order $2$, then the order of the group is divisible by $8$

Prove that if $|G| < \infty$ and $|G'| = 2$ then $|G|$ is divisible by $8$. Thoughts. $A \simeq G / G'$ is abelian and $G' \simeq \mathbb{Z}_2$. Since $G' \subset G$ then at least $|G| \vdots 2$. ...
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1answer
22 views

Vector space generated

Let $(L,+)<(\mathbb R^n,+)$ be a additive subgroup and let $\{v_1,\ldots,v_m\}$ be a maximal linearly independent subset of $L$. Let $V$ be the subspace spanned by $\{v_1,\ldots,v_m\}$. Asumme that ...
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1answer
37 views

How to prove there are only finitely many distinct groups with exactly n elements?

How to prove there are only finitely many distinct groups with exactly n elements, If isomorphic groups are regarded as being the same? I set $f: a_{ij}\to b_{ij}$, then it is easily to prove $f$ is ...
2
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0answers
33 views

Generators of group of “unitary” matrices over a finite field

This is about a group related to $U(n,q)$ and $SU(n,q)$. I know from multiple sources the generators for these groups, but $U(n,q)$ is defined to be the group of matrices $A$ such that $A^*JA = J$ ...
1
vote
1answer
25 views

Unique image of torsion groups in the circle group.

Let $p$ be a prime. For any one-to-one homomorphisms $f,g:\Bbb Z_{p^\infty}\to \Bbb T$, we have $f[\Bbb Z_{p^\infty}]=g[\Bbb Z_{p^\infty}]$, where $\Bbb T $ is the circle group. Is this correct ...
2
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2answers
40 views

Clarification on Symmetry Group

My text says "It is a general fact, and an easy one to prove, that the invertible transformations of a mathematical object that preserve some feature of its structure always form a group. We call ...
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2answers
80 views

Factor sets and group extensions (Homological algebra- Hilton and Stammbach VI.10.1)

Show that an extension $$A\xrightarrow{i} E\xrightarrow{p} G$$ may be described by a factor set, as follows. Let $s:G\rightarrow E$ be a secion so that $ps=1_G$. Every elmenet of $E$ is of the form ...
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3answers
33 views

symmetric difference

I have to prove that the symmetric difference, AΔB = (A∪B) \ (A∩B), is associative for my mathematics study with two inclusions. So I have to prove that: (AΔB)ΔC = AΔ(BΔC) I started with taking an ...
5
votes
1answer
63 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism and that $f_* : H_2(G)\rightarrow H_2(H)$ is an epimorphism. Question is to prove that this ...
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3answers
38 views

Group homomorphism that maps to one element

Is there some way to construct a group homomorphism $G \to H$ that maps everything in $G$ to just one non-identity element of $H$ (besides mapping $0$ to $0$)? For arbitrary (finite) $H$ but you have ...
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0answers
20 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
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1answer
47 views

Isomorphisms based on Conjugacy Classes

For what groups of the same order are not isomorphic and contain the same conjugacy class? I as well have a more detailed question: For which of those groups are not abelian. The only example I know ...
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3answers
174 views

$|G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + \frac{|G|}{\left|\langle ab\rangle\right|}$

Show that for every finite group $G$ and for every elements $a, b \in G$ the following expression $$ |G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + ...
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3answers
37 views

What is a conjugacy class of reflection?

I have a problem to do, asking to show that $D_{2n}$ has two conjugacy classes of reflections if n is even, but only one if n is odd. My question is, what is a conjugacy class of reflection? I have ...
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2answers
41 views

Show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$

I am asked to show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$, where $G'$ is the commutator subgroup of $G$, and $C :=\{aba^{-1}b^{-1}\mid a,b\in G\}$. Showing $\bigcap_{C\subseteq N ...
3
votes
2answers
58 views

Groups of Order $n$

Is there a formula for finding the number of groups of order n. For example, if a group $G$ has an order n, is there a formula in which someone can find the number of groups with that order. I suppose ...
2
votes
2answers
32 views

Prove that the group $\mathbb{Z}^{n}$ is generated by at least $n$ elements

I need to prove that the group $\mathbb{Z}^{n}$ with the regular $+$ operation is generated by at least $n$ elements. I know it's pretty analog to the case of vector spaces.. I tried induction ...
-3
votes
1answer
38 views

A Problem Involving Isomorphism . [closed]

Please help me with this.. A group theoretic proof that $(\mathbb Q,+)$ is not isomorphic to $(\mathbb R^+,*)$.??