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

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Conjugation and simple group

I'd like to prove group $G$ is simple if and only if for every pair $g,h \in G$ ($g \ne 1$), $h$ can always be written product of finite number conjugates of $g$ and $g^{-1}$.
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384 views

Open subgroup of $SO(3)$

Does $SO(3)$ have an open nontrivial subgroup?(Group $SO(3)$ with usual matrices product, is all $3\times 3$ matrices whose determinant is 1 and for every element $A\in SO(3)$ we have $A^tA=AA^t=I_3$ ...
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117 views

On nilpotent factor group

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ with the property that $G/N$ is nilpotent. Prove that there exists a nilpotent subgroup $H$ of $G$ satisfying $G = HN$. This is ...
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135 views

At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic [closed]

Suppose we have a finite group $G$ of finite order $n$. For every $d\mid n$, $G$ has at most one subgroup of order $d$. Show that $G$ is cyclic.
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71 views

About using Burnside's Theorem

The question is the following. Consider a square toy made out of 4 balls on the corners connected by 4 symmetric pipes. How many distinct (up to rotation and flipping) ways are there to make ...
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35 views

a exercise in Berkovich‘ book

in Berkovich' book characters of finite groups there is a exercise in page 59. exercise 15, a group G is a Q-group if and only if for any cyclic subgroup Z of G who can tell me how to prove it ? ...
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92 views

what is $\mathbb{Q}/\mathbb{Z}$ isomorphic to?

I'm trying to classify it using the fundamental homomorphism theorem. Any hints? Is this quotient a familiar group?
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60 views

An element of order $n$ generates a normal subgroup of $D_n$

Let $a$ be an element of order $n$ of $D_n$. Show that $\langle a\rangle \lhd D_n$ and $D_n/\langle a\rangle \cong \mathbb Z_2$. Proof: Let $K = <a>$ for some a ∈ G. Let H ≤ K be an ...
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80 views

Subgroup of a Direct Product

Let $G$ and $H$ be groups and $G\times H$ their direct product. a) Prove that $\{(x,e) : x\in G\}$ is a subgroup of $G\times H$ b) Prove that $\{(x,x) : x\in G\}$ is a subgroup of $G\times G$ I ...
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82 views

Cyclics groups and Lagrange Theorem

Let $G=\langle y\rangle$ be a cyclic group of order $n$. Prove that $\langle y^k\rangle \subseteq\langle y^m\rangle $ if and only if $\gcd(n,m)$ divides $\gcd(n,k)$. So I think I was able to ...
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48 views

Non-isomorphic product of two groups

I know this is a simple question, but I'm not able to reason it out right now. Why is $\pm I_n \not\cong \pm I_n \times \pm I_n$?
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47 views

Understanding the quotient of infinite groups $\mathbb{R}^2/H$ where $H = \{(a, 0): a\in \mathbb{R}\}$

Define $H = \{(a, 0): a\in \mathbb{R}\}$. Without using the fundamental homomorphism theorem, how would we know what $\mathbb{R}^2/H$ is? The quotient group is $\{H, (x, y) + H, (x_2, y_2) + H, \dots ...
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47 views

Generators of Groups

I need to show the following: Show that $\mathbb{Z}$ is generated by $5$ and $7$. I think that the solution has to do with relative prime numbers but I don't know where to start.
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51 views

Commutator subgroup [duplicate]

Assume that G is a group. Define $G'=\langle\{ghg^{-1}h^{-1}\mid g,h\in G \}\rangle$. Then $G'$ is a normal subgroup of $G$. Prove If $H$ is a subgroup of $G$ and $G'\subseteq H$, then $H$ is normal ...
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34 views

determining abelian groups of a certain size up to isomorphism

Say the size is 360. My book uses this as an example. It says there are 6 distinct groups up to isomorphism: $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times ...
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2answers
192 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
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261 views

Factor Group over Center Isomorphic to Inner Automorphism Group

I'm having some trouble proving that for a group $G$, $G/Z(G)\cong\text{Inn}(G)$, where $Z(G)$ is the center of the group defined as $Z(G)=\{z\in G:gz=zg\forall g\in G\}$ and $\text{Inn}(G)$ is the ...
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124 views

Elements of order $10$ in $\Bbb Z_2 \times \Bbb Z_{10}$

How many elements in the group $\mathbb Z_2 \times \mathbb Z_{10}$ are of order $10$? I think the easiest way to answer this question might be to write them out, but I'm not sure how to write ...
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92 views

Stabilizer map on transitive G-set defines a morphism with G acting on subgroups by conjugation

This is part of a homework problem for a graduate course on abstract algebra. Given a transitive G-set $X$, show that the map that assigns to $x \in X$ its stabilizer defines a morphism of G-sets ...
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1answer
88 views

Can you give me a good alternative to Rotman's Group Theory book?

I've been trying to learn out of Rotman's book "An Introduction To The Theory of Groups" for the last few months, and it's rough going. I've been studying Chapters 7, 10, and 11 in particular, and ...
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59 views

A subgroup of $S_n$ of odd order is contained in $A_n$

I saw this question. The original questioner asserted that the Lagrange's Theorem is sufficient to solve the problem, but I think that the theorem does just say that the order of $H$ divides the order ...
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3answers
191 views

Decomposition of a cycle as a product of transpositions

Can someone please explain the rules pertaining to different ways to write a cycle decomposition as products of 2-cycles, an example from textbook: I understand this $$ (12345) = (54)(53)(52)(51) ...
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1answer
45 views

For what n does $|A|=n, \ A \in SL_2(\mathbb{Z})$?

Could you help me check for what n does $|A|=n, \ A \in SL_2(\mathbb{Z})$? I know that given two eigenvalues $\alpha, \beta$ of a $2 \times 2$ matrix $A$, $A^n = \alpha ^n (\frac{A-\beta I}{\alpha - ...
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66 views

What does “nilpotent” in a “nilpotent group” mean?

It seems to have nothing to do with the usual nilpotency, i.e. $\exists n\in \mathbb{N}:x^n=0$. Actually I think the latter only makes sense in a ring or more rich structure. I tried to relate some ...
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181 views

Left Cosets and Right Cosets.

Recall that $GL_2(\mathbb{R})$ is the group of all invertible 2x2 matrices with real entries. Let: $G = (\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \in GL_2(\mathbb{R}) : ac \neq 0$) and ...
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170 views

List all cosets of H and K

Let $G = \mathbb Z_3 \times \mathbb Z_6$, $H = \langle (1,2)\rangle$ and let $K = \langle (1,3)\rangle$. List all cosets of $H$ and $K$. Can somebody please explain me how to do this problem. ...
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96 views

group generators of $(\mathbb Z_{17}-\{0\},\times)$

How to find generators of $(\mathbb Z_{17}-\{0\},\times)$? Is there a faster way to find generators than trying every element in the group? I know that for additive group, if a number say m is ...
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53 views

Order of elements of a group.

Assume that G is an abelian group, and a∈G. (a) Assume that |a|=r and that m|r, say r=mt. Prove that $|a^t|$=m. Proof:Assume a∈G and |a|=r and m|r, say r=mt. Assume $|a^t|$=k Since ...
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Confused by Example in Herstein's “Topics in Algebra”

The following comes from I.N. Herstein's "Topics in Algebra", just after defining subgroups. He gives the following example Let $S$ be any set and $A(S)$ be the set of one-to-one mappings of $S$ ...
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141 views

Infinite coproduct of abelian groups

One can see on every text (book, lesson, comments) that a direct sum/coproduct of abelian groups is the same as a finite product but in the infinite case, the direct sum/coproduct is only a subgroup ...
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63 views

I don't think I'm using an assumption in this proof. Anything wrong?

Define the exponent $\exp(G)$ of a finite group $G$ to be the smallest positive integer $k$ such that $g^k = e$ for all $g \in G$. The question asks If $G$ is a finite abelian group, prove that ...
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26 views

About a finite group of lie type

According to wikipedia, the order of $^2A_n(q^2)$ if $$\frac{1}{(n+1,q+1)} q^{n(n+1)/2} \prod_{i=1}^n (q^{i+1} -(-1)^{i+1}).$$ So when $n=1,q=2$, this order is $6$. Let $F = \{0,1,a,a+1\},$ ...
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75 views

What can conclude about $[G:H]$?

Assume that $H$ is a subgroup of a finite group $G$, and that $G$ contains elements $a_1, a_2,...,a_n$ such that $a_i a_j^{-1} \notin H $ for $1\leq i < n, 1 \leq j <n $, and $i \neq j$. What ...
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57 views

Find the number of cosets$ [G:H] $?

Assume that $G$ is a cyclic group of order $n$, that $G =\ <a> $, that $k|n$ , and that $H=<a^k>$. Find $[G:H] $ the number of cosets to the subgroup H I think that since $k|n$ ...
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250 views

Groups : $a$ and $b$ commute, prove $a^2$ commutes with $b^2$

If $a$ and $b$ are in $G$ and $ab=ba$ we say that $a$ and $b$ commute. Assuming that $a$ and $b$ commute prove the following: 1) $a^2$ commutes with $b^2$
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105 views

Euler function and $\mathbb{Z}/n\mathbb{Z}$

I am trying to solve a very interesting problem about the ring $\mathbb{Z}/n\mathbb{Z}$ and Euler function $\phi (n)$, but i am not sure how to start, i have a few ideas, but none of them leads me to ...
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109 views

Finding the smallest set on which a group acts faithfully

Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the ...
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49 views

Proving if $H<G$ and $N\triangleleft$ G then $H\cap N \triangleleft H$

I want to prove that if $H<G$ and $N\triangleleft$ G then $H\cap N \triangleleft H$. Here is what I have done: $N\triangleleft G \iff aN=Na \quad \forall a\in G$ $hN=Nh \quad \forall h\in H$ ...
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70 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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132 views

How many elements are there in the intersection of two subgroups of a finite cyclic group?

Let's assume that we have two subgroups $H_1$ and $H_2$ in $\mathbb{Z}_n$ with $k$ and $l$ elements respectively. How many elements are there in the intersection $H_1\cap H_2$? Let denote this by ...
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375 views

Is it possible to have a non-trivial homomorphism of some finite group into some infinite group?

Let $G$ be a group of some finite order, and let $G^\prime$ be some group of infinite order. Then there is the trivial homomorphism of $G$ into $G^\prime$ which maps each element of $G$ into the ...
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125 views

Why is the group of units mod 8 isomorpic to the Klein 4 group?

I recently learned that $U_8\cong \mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$. I can see, through a bit of computation, that this is the case, but I was wondering if this is just a coincidence ...
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152 views

When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?

Let $m$ and $n$ be two given positive integers such that $m<n$. Then what are the necessary and sufficient conditions for the groups $(\mathbb Z_m,+_m)$ and $(\mathbb Z_n,+_n)$ to be homomorphic ...
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95 views

Determining the automorphism group of a disconnected graph

There is this know formula for determining the automorphism group of a graph $G$: let the connected components of $G$ consist of $n_1$ copies of $G_1$, $\dots$, $n_r$ copies of $G_r$, where $G_1, ...
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If a group $G$ has odd order, then the square function is injective.

Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this ...
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95 views

$GL_n(F)$ is not abelian for $n\ge 2$ counter example?

So I am asked to prove that for $n \ge 2$, the group $GL_n(F)$, where $F$ is any field, is non-abelian. I figure this amounts to finding a counter-example for all such $n$. It wasn't hard, but I'm ...
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113 views

Is a congruence (equivalence) class modulo n a group?

I know that the set of all equivalence classes Z/nZ is a group (with identity element the equivalence class [0], inverse element -[a]=[n-a]=[-a], etc.). However, is a single equivalence class modulo ...
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54 views

Show that HK is a subgroup of G.

Hello stackexchange mathematicians, So I am stuck at (a), what is it asking me? (if and only if $KH \subset HK$) Also what does $|G:H|$ and $\langle H \cup K\rangle$ notation mean?
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185 views

Show there exists a fixed positive integer $n$ such that $a^n = e$ for all $a\in G$.

Let $G$ be a finite group. Show there exists a fixed positive integer $n$ such that $a^n = e$ for all $a\in G$. We know: $n$ is independent of $a$.
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55 views

What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...