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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How Many Orbits Does The Symetric group action Sym(6) on itself have

Let the group action $g \hookrightarrow X$ Let's define the following terms: orbit: Let $x \in X$ it's orbit is the set $G.x =${$g.x | g \in G$}$\in X$ Sym(X): It's the set of bijections $X ...
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14 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
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1answer
14 views

If the order of $G$ is odd, show that for any $y \in G$ there is a unique $x \in G$ such that $x^2=y$

This question seems to be a typical one, but I am not sure about the uniqueness part. My attempt Suppose $|G|$ is odd. By Lagrange's Theorem, we have all elements in $G$ is of odd order. In ...
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1answer
17 views

General property of Fitting series

Let $G$ be a finite solvable group. $F(G)$ (Fitting subgroup) is defined to largest normal nilpotent group contained in $G$. Then $F_2(G)$ is defined to be inverse image of $F(G/F(G))$. i.e ...
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What is $gnu(18,480)\ $?

Probably, the number of groups of order $18,480$ can only be determined with GAP. But I may be wrong, so the question should not be understood to be purely computational. A better lower bound than my ...
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1answer
20 views

Prove that $a . g = g^{-1}.a$ and $g . a = a . g^{-1}$ hold for any group and any action defined.

Dummit and Foote page 129 claims that for arbitrary group actions that if we are given a left group action of $G$ on $A$ then the map $A \times G \to A$ defined by $a . g = g^{-1}.a$ a is a right ...
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1answer
26 views

Quotient involving $\pi$-subgroups

Let $G$ is a finite group and $\pi$ be a set of primes. Suppose that $P$ is a normal $\pi$-subgroup. Is it true that the quotient group $G/P$ is $\pi$-group? I know that since $P$ is a ...
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8 views

Geometrical interpretation of the conjugacy of triangle groups.

Let $\triangle$ and $\triangle'$ be two hyperbolic triangles of respective angles $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$. Let us suppose that the triangle subgroups ...
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49 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
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1answer
70 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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Quotient ring $(\mathbb{Z}_4 \times \mathbb{Z}_6)/S$

Consider the ring Z4xZ6 with +6, *6, and +4, *4 in appropriate coordinates and S={(0,0),(2,0),(0,3),(2,3)}. Would the elements of the quotient ring Z4 x Z6 / S be: S+0 (trivial set above), ...
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0answers
24 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
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1answer
29 views

what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

I am looking for smallest example of a group $G$ such that: $G$ is a finite, non-abelian group $G$ is not simple $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $ $H_1, H_2, \dots $ ...
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43 views

Is the size of the conjugacy class of a given element in a compact Lie group always finite?

Let $G$ be a compact Lie group and $g\in G$ be any given element in it. Consider the conjugacy class of $g$ in $G$, denoted by $[g]=\{hgh^{-1}:h\in G\}$. Our question is that: Could you find a ...
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1answer
18 views

Prove that for any n in $ \Bbb N $ there exists a unique cyclic subgroup $H_n$ $\subset$ G of order n. Where G is the group $\Bbb Q/$ $\Bbb Z$.

Prove that for any n in $ \Bbb N $ there exists a unique cyclic subgroup $H_n$ $\subset$ G of order n. Where G is the group $\Bbb Q/$$\Bbb Z$. Existence was not difficult to show: Let $H_n$ = < ...
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1answer
31 views

Does this also define a normal subgroup?

The definition of a normal subgroup is: $H$ is said to be a normal subgroup of $G$ if: $h \in H$, and $g \in G$ $\implies$ $ghg^{-1} \in H$ I am wondering if the following also defines a normal ...
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1answer
26 views

Describe the orbits of the action.

So $L$ denotes the set of oriented straight lines through the origin in $\mathbb{R}^{2}$ (that is, straight lines with a preferred direction, indicated by an arrow). The group $(\mathbb{R},+)\cong ...
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1answer
25 views

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
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1answer
26 views

number of generators of a semi direct product

Let $G$ be a finite group. Let $g(G)$ be the minimum set of elements of $G$ required to generate the whole group. Suppose that $G= H \rtimes K$ is a semi direct product of two finitely generated ...
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2answers
53 views

Does $(\mathbb{Z} \times \mathbb{Q})/M$ have any element of infinite order?

Let $\mathbb{Z} \times \mathbb{Q}$ be the group of ordered pairs $(x, y)$ with $x \in \mathbb{Z}, y \in \mathbb{Q}$ under component-wise addition. Fix $m \in \mathbb{Q}$ and let $M \subset \mathbb{Z} ...
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39 views

Equality of two functions.

I have a specific question, from a paper given below. Here I got an answer of question: When two functions are called equivalent?.It helped me to understand the first and the second steps of the ...
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121 views

There does not exist group $G$ such that Aut$(G)\cong \mathbb{Z}_n$ (for odd $n$)

I had this "almost bonus" question on the final in Group Theory recently: prove that there is no such group $G$ which would satisfy Aut$(G)\cong \mathbb{Z}_n$, where $n$ is an odd integer. I don't ...
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32 views

For all $g\in G$, is it true that $gHg^{-1} \subseteq H $ if H is a normal subgroup of G.

Assume that $H$ is a normal subgroup of group $G$. Is it true that for all $g∈G$ one has $gHg^{-1} \subseteq H$ ? I know that if H is a normal subgroup of G, then $ghg^{-1}$∈ $H$ where $h$∈$H$ and ...
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1answer
44 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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2answers
892 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does "$G$ is abelian" or "$G$ is cyclic" imply "$\text{Aut }G$ is cyclic"?
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Give me your opinion about those books [on hold]

I found some books printed, and I want somebody to tell me their opinion about them for general reading (about their level, difficulty, worth reading etc), not just for a particular course in ...
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isomorphic permutation groups with same cycle index [on hold]

Is there two nonidentical isomorphic permutation groups with same cycle index?
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+50

Matrix representations of the generators of the full octahedral group

I want to find matrix representations of the generators of the full octahedral group which has the presentation $\{a,b,c|a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\} $ where a,b and c are the generators of the ...
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0answers
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Generating $\mathrm{SL}(n,\mathbb{R})$

I have this question which doesn't seem too difficult and I would like to know if there is an elementary way to deal with it. I consider a closed subgroup $G$ of $\mathrm{SL}(n,\mathbb{R})$ which ...
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2answers
258 views

What's the automorphism group of the real and complex numbers and the quaternions?

According to Wikipedia the automorphism group of the octonions is the exceptional group $G_2$. Are there analogous groups for the real numbers, the complex numbers and the quaternions?
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1answer
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$G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?

Let $G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ? Since $Aut(G)$ is cyclic here , I know that $G$ is abelian , but this is as far as I can get . Please help . Thanks in ...
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0answers
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why if $\frac{G} {Z(G)} \cong Z_P. Z_P$ then $G' \cong Z_P$? [on hold]

why if $\frac{G} {Z(G)} \cong Z_P. Z_P$ then $G' \cong Z_P$? can you help me?
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Triangle group $(\beta,\beta,\gamma)$ is a subgroup of the triangle group $(2,\beta, 2\gamma)$.

Let $1/\alpha+1/\beta+1/\gamma<1$, and let us consider the triangle group $(\alpha,\beta,\gamma)$, i.e. the subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ induced by the hyperbolic triangle ...
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1answer
59 views

When is this cyclic representation irreducible?

Let $G$ be a finite group, and let $(\rho, W)$ be a representation of $G$ on $W$. We assume that $W = \bigoplus_i W_i$ is a direct sum of equivalent irreducible representations $W_i$. There are many ...
2
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1answer
32 views

$G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic?

If $G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic , or equivalently asking , then is $[G:H]$ finite for every non trivial ...
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1answer
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Chain Rule of Calculus as a Group Property?

I read that the chain rule and inverse function theorem are expressions of the group property of successive non-singular transformations. How do you say this more formally? My guess is that we are ...
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Proving a conjugation map is an Inner automorphism of a group

Definition: The map $i_{g}:G\rightarrow G$ $h\mapsto g^{-1}hg$ Lemma: $i_{g}$ is an Automorphism of G called an Inner Automorphism. My attempt to prove this is as follows: ...
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28 views

Show that $B$ is an infinite group. [on hold]

So $B$ denotes the group with presentation $\langle s,t\mid stst^{-1}s^{-1}t^{-1}\rangle$. What is a good way to prove this? Thanks a lot.
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1answer
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Prove that a semigroup which satisfies a certain conditions is a group [on hold]

This is an exercise from "Abstract Algebra" by P.A.Grillet (p.12, ex.2). Let $S$ be a semigroup (that is, a set with an associative binary operation) in which there is a left identity element ...
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Prove neither $F_4$ nor $F_5$ is isomorphic to $\mathbb{Z}^4$?

Prove neither $F_4$ nor $F_5$ is isomorphic to $\mathbb{Z}^4$. ($F_4,F_5$ are free groups.) My first thought is to show that $F_4$ and $F_5$ are not free abelian groups and thus cannot be ...
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H prime order, normal subgroup of group G. Prove H in center Z(G).

I am looking at the following question: "Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime that divides the order of G. Prove that H is in the ...
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33 views

Find the orbit of $x$

Let $G=D_{2n}$ and let $g\dot{}x=gxg^{-1}$ define and action of $G$ on itself. Find the orbit and stabilizer if $x=a.$ Note: Here $a$ denotes a rotation through $\frac{2\pi}{n}$ and $b$ denotes a ...
2
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1answer
80 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
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1answer
26 views

Orbit and stabilizer

Let $G$ be a group and let $g \dot{}x=gxg^{-1}$ for all $g,x\in G.$ Find the orbit and stabilizer when $x=e.$ Orbit: $$G\dot{}x=\{g\dot{}x \ \colon g\in G\}=\{g\dot{}e \ \colon g\in G\}=G$$ ...
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Prove that if $H$ is an abelian subgroup of a group $G$ then $\langle H, Z(G)\rangle$ is abelian.

Prove that if $H$ is an abelian subgroup of a group G then $\langle H, Z(G)\rangle$ is abelian. So I started out trying to show that $H$ is a subset of $Z(G)$ but then I realised that $Z(G)$ is the ...
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0answers
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about minimal non-nilpotent groups

Newman and Wiegold have studied the AN-groups i.e. the locally nilpotent groups which are not nilpotent but every proper subgroup is nilpotent. I was asking why the notion locally nilpotent was added ...
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4answers
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Why is the 'Law of Cancellation' for groups only an implication?

It is easy to see that for a Group $G$ and $a,b \in G$ $ab = ac \Rightarrow b = c$ (See also here) But what is about the other direction? That is: $b = c \Rightarrow ab = ac$ Does this ...
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2answers
22 views

Definition of subgroup of abelian group $G$ generated by subset $A$

In my book I have the following definition for subgroups of a group $G$ generated by $A$, a subset of G: $$\langle A\rangle=\{x_1^{\epsilon_1}x_2^{\epsilon_2}...x_n^{\epsilon_n}\mid x_i\in ...
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3answers
47 views

Group of order $p^2$. [closed]

$G$ is a group of order $p^2$, $p$ is a prime. If $Z(G)=p$ then $G/Z(G)$ has order $p$ and $G$ is cyclic. Why $G$ is cyclic? Is it related to the Lagrange Theorem? Actually, I have no idea about it. ...
2
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1answer
61 views

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$.

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$. I think this is associated with the action of the left coset of ...