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

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20 views

I don't understand this notation- abelian groups

May be a stupid question but is $(\mathbb{Z}^n)_p \equiv \mathbb{Z}^n/(\mathbb{Z}p)^n$ (when $p$ is a prime)??
6
votes
2answers
93 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
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1answer
89 views

How to find out whether a group is Abelian

Let $G$ be the set $\mathbb R\setminus \{0\}$ and let $*$ denote a binary operation on the set, defined by:$$\forall a, b \in G,\;\;\;a*b= \dfrac{a\cdot b}2.$$ I need to show that $[G,*]$ is an ...
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0answers
8 views

herstein excercise on a finite group

I'm stuck on this herstein exercise for a long time. Let $P$ is a $p$-Sylow subgroup of $G$ and order of $a$ is a prime power then if $a\in N(P)$ prove $a\in P$ I was doing like this but stuck in ...
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1answer
25 views

Prime numbers $p$ and $q$ and possession of normal subgroup of order $p$

I have this following question from my class note on Sylow Theorem: Let $p$ and $q$ be prime numbers such that $p \nmid (q-1).$ Show that each group of order $pq$ possesses a normal subgroup of ...
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1answer
23 views

homomorphism/ isomorphism

Let $f : G \to H$ be a homomorphism of groups. Let $K$ be a subgroup of $H$, and $A$ a subgroup of $G$. Show that (1) $f^{-1}(K)$ is a subgroup of $G$, (2) $f(A)$ is a subgroup of $H$, (3) if $G$ is ...
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0answers
10 views

Mobius transformations — $L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$

$$L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$$ I cannot seem to get the right answer when I multiple them out. $$(1-i)z \cdot (1-z) \cdot \frac{z}{1-i}$$ What do you get when you multiple ...
4
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1answer
38 views

Divisibility of group exponents when the subgroup has finite index.

Let $G$ be a group (not necessary finite) and $H$ a subgroup of $G$ of index $n$ such that exp $(H)<+\infty$ . Show that $$\exp(G)<+\infty$$ and $$\exp(G)\mid\exp(H)\cdot n.$$ Remarks. ...
15
votes
2answers
658 views

Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?

This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that: $L_4(2)$ and $L_3(4)$ both have order $20160$ $O_{2n+1}(q)$ and $S_{2n}(q)$ ...
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1answer
126 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
5
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1answer
51 views

Tensor product of (general?) groups

I am starting to learn about tensor products of abelian groups. Why is the tensor product defined for abelian groups? In which part of the construction the commutativity of the groups is needed?
2
votes
1answer
35 views

Is my proof correct? ($A_n$ is generated by the set of all 3-cycles for $n \geq 3$)

I want to prove that for $n \geq 3$, the alternating group $A_n$ is generated by the set of all 3-cycles. Here is my attempt: Let $\mathcal{S}$ be the set of all 3-cycles in $S_n$, which is a ...
0
votes
0answers
30 views

finite index subgroups of profinite completions

Let $G$ be a finitely generated, residually finite group, and let $\widehat{G}$ denote its profinite completion. Is there a 1-1 correspondence between finite index subgroups of $G$ and open subgroups ...
3
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2answers
75 views

Group of order 30 can't be simple

I have this following question from my class note on Sylow Theorem: Show that a group of order 30 can not be simple. For that I know the followings: (1) A simple group is one that does not have ...
4
votes
0answers
61 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
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1answer
27 views

What does “exponent 2 nilpotency class 2” mean?

According to the book The Symmetries of Things, p. 208, the number of groups of order 2048 "strictly exceeds 1,774,274,116,992,170, which is the exact number of groups of order 2048 that have ...
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6answers
203 views

Abelian group of order 99 has a subgroup of order 9

Prove that an abelian group $G$ of order 99 has a subgroup of order 9. I have to prove this, without using Cauchy theorem. I know every basic fact about the order of a group. I've distinguished ...
2
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1answer
25 views

Minimal number of relations in finite 2-groups with 2, 3, and 4 generators

I have received helpful answers to my two previous questions that focused on the symmetric group of degree 3 and the dihedral group of order 8. If $d$ is the minimal number of generators of a finite ...
3
votes
2answers
303 views

If consecutive elements commute each other, does it mean that all of them commutes with each other?

Let $x_1,x_2,...,x_k$ be $k$ different elements of a group $G$ and $k\geq4$. If we know that $x_i$ commutes with $x_{i+1}$ and $x_k$ commutes with $x_1$, can we say that all $x_i$ commutes with each ...
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1answer
30 views

Conditions for finiteness of group in geometric group theory

Are there any sufficient conditions in geometric group theory for a group to be finite? Are there any necessary conditions?
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2answers
27 views

Problems about generators for Sylow p-subgroups

There are several problems I met asking to find the generators for some different Sylow $p$-subgroup. $(i)$ a Sylow 2-subgroup in $S_{8}$; $(ii)$ a Sylow 3-subgroup in $S_{9}$; $(iii)$ a Sylow ...
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vote
1answer
19 views

Sketch a figure which has a group of symmetries of order 5.

I am trying to draw a shape which has only 5 symmetries I know Square has 8 Rectangle/parallelogram has 4 Triangle has 6 Circle has infinite how do i know which shape has only 5 I know that ...
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2answers
158 views

Dihedral Group Computations

Let $n > 1$ be an integer and let $\theta = \dfrac{2\pi}{n}$. Let $P$ be the regular $n$-gon with vertices ($\cos i\theta$, $\sin i \theta$) for $i \in \mathbb Z_n$. The dihedral group $D_n$ is the ...
2
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0answers
61 views

free group with 2 generators (two matrices)

Let $\alpha$ be complex number such that $| \alpha | > 1$. Show that $\left(\begin{array}{cc}1&0\\\alpha&1\end{array}\right) $ and ...
4
votes
1answer
53 views

Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
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2answers
18 views

Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
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0answers
28 views

$\langle g\rangle$ is $p$-Sylow subgroup of $G_\Delta$?

Let $p$ be a prime and $G$ a primitive group of degree $n=p+k$ with $k\geq3$. If $G$ contains an element of degree and order $p$. $G$ contains the cycle $(1,2 ... p)=g$. Let $\Delta= \lbrace ...
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0answers
28 views

prove that $O^{\pi}(G) \leq K$ . [on hold]

Suppose $G$ is a finite group and $\pi$ will be a set of prime numbers (not empty), if $K \triangleleft \triangleleft G $ ($K$ is subnormal in $G$ ) and $[G:K]$ is a $\pi$-number, then $O^{\pi}(G) ...
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1answer
38 views

Quick question: G-set functor

The Wikipedia page on Representable Functor says: A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a ...
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2answers
189 views

What is a Simple Group?

I am working on this problem from class note: Let $G'$ be a group and let $\phi$ be a homomorphism from $G$ to $G'$. Assume that $G$ is simple, that $|G| \neq 2$, and that $G'$ has a normal ...
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3answers
70 views

Can we conclude that $A= B$?

Let $G$ be a group. Suppose that $A\leq B\leq G$ and $[G,A]= [G,B]$. Can we conclude that $A= B$ ?
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2answers
49 views

Group Transitive Action's Effect on Stabilizers's Conjugacy

I am looking for guidance for two problems on group action, one of them is here and the other one has just been posted earlier: Assume that $G$ operates on a set $\Omega.$ Show that, if $G$ acts ...
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3answers
113 views

Two problems in Group theorem related to Sylow's theorem(maybe)

Prove that any subgroup of order $ p^{n-1} $ in a group $G$ of order $p^{n}$, p a prime number, is normal in $G$. $(a)$ Prove that a group of order 28 has a normal subgroup of order 7. To deal ...
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0answers
55 views

For a simple nonabelian group every automorphism with $xf(x)=f(x)x$ is trivial.

Suppose that $G$ is a simple nonabelian group. Prove that if $f$ is an automorphism of $G$ such that $xf(x) = f(x)x$ for every $x\in G$, then $f = 1$.
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1answer
136 views

If $G/G'$ is finite, then $|Z(G)| < \infty$

Let $G$ be an infinite group. Suppose that $G/G^{\prime}$ is a finite group, where $$G^{\prime}= \left\langle xyx^{-1}y^{-1}\ \middle\vert\ x,y \in G\right\rangle.$$ Prove that $|Z(G)| < \infty$.
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1answer
425 views

What is $\operatorname{Aut}(\mathbb{R},+)$?

I was solving some exercises about automorphisms. I was able to show that $\operatorname{Aut}(\mathbb{Q},+)$ is isomorphic to $\mathbb{Q}^{\times}$. The isomorphism is given by $\Psi(f)=f(1)$, but ...
5
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1answer
45 views

Proving that if the semigroup (A, *) is a group, then the relation is an equivalence relation.

I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this: Throughout this ...
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votes
3answers
69 views

Can $\mathbb{Z}/n\mathbb{Z}$ (not $(\mathbb{Z}/n\mathbb{Z})^{\times}$) be a group under multiplication?

I was wondering why we usually say $\mathbb{Z}/n\mathbb{Z}$ is a group under addition and invent notation like $(\mathbb{Z}/n\mathbb{Z})^\times$ specifically for the multiplicative group modulo $n$. ...
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votes
4answers
2k views

Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.

Can anyone please help me out on how to use the fundamental theorem of finitely generated abelian groups to prove that $({\mathbb{Q}},+)$ is not finitely generated?
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0answers
31 views

Sylow subgroup of $S_{11}$

I want to construct some Sylow $3$-subgroup of $S_{11}$.This subgroup has $3^4$ elements. I know any Sylow $3$-subgroup is isomorphic to $(\mathbb{Z}/3\mathbb{Z})^3\rtimes P$ where $P$ is a Sylow ...
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1answer
28 views

Injective homomorphism between a finite group $G$ and $GL_n(\mathbb{F}_p)$ where $p$ is prime

I'm looking for a solution to the following problem: Given a natural number $n$, a prime number $p$ and a finite group $G$, I need to find an injective homomorphism between $G$ and the group ...
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3answers
110 views

Group of order $p^2$ is commutative with prime $p$

Please help me on this one: Let $p$ be a prime number, show that each group of order $p^2$ is commutative. If you do not mind at all, could you please not give me the elegant explanation, but ...
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1answer
27 views

Isomorphism of quotient of direct sum modules

Let $M, N, M'$ and $N'$ be R-modules. If $M'$ and $N'$ are submodules of both $M$ and $N$ then is it true that \begin{equation} \frac{M}{M'} \oplus \frac{N}{N'} \cong \frac{M \oplus N}{M' \oplus N'} ...
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1answer
38 views

Prove that (G,*) is a group.

G is a monoid and satisfies the right inverse law. Show that G is a group. I tried next: It is obviously sufficient to show that G satisfies left inverse law. (I use $a'$ notation for $a$ inverse.) We ...
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0answers
25 views

I can't prove theorem 13.9 on finite permutation groups of Wielandt book.

I can't prove this theorem on finite permutation groups of Wielandt book. Theorem 13.9: Let $p$ be a prime and $G$ a primitive group of degree $n=p+k$ with $k\geq3$. if $G$ contains an element of ...
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1answer
64 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
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1answer
34 views

Subgroups and an union of orbits

I have to prove or disprove the following statement: If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an ...
2
votes
1answer
46 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
2
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2answers
48 views

Group theory, the squares of G

We have a group $G$ with a subgroup $G_2$, which is defined by $G_2:=\{g^2|g \in G \}$. I have to prove that i) $G_2\triangleleft G$ ii) all elements of $G/G_2$ have order $\leq2$ iii)if ...
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0answers
29 views

Bounds on the numbe of groups of degree n [duplicate]

What are the best lower/upper bounds on the number of (non-isomorphic) arbitrary groups of degree n? Thanks!