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

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5
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1answer
40 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?
7
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1answer
106 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 ...
3
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2answers
62 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 ...
1
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1answer
17 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 of order $pq$ possesses a normal subgroup of oder ...
4
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0answers
50 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 ...
2
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1answer
32 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
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1answer
22 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 ...
0
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0answers
25 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 ...
1
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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 ...
3
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2answers
180 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 ...
2
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2answers
26 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 ...
4
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6answers
196 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 ...
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2answers
15 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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votes
0answers
26 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 ...
0
votes
1answer
34 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 ...
0
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0answers
26 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) ...
0
votes
3answers
67 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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0answers
51 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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3answers
108 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 ...
4
votes
1answer
43 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 ...
4
votes
1answer
52 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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0answers
29 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 ...
1
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1answer
26 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 ...
0
votes
1answer
26 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'} ...
2
votes
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 ...
2
votes
0answers
59 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 ...
0
votes
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 ...
2
votes
1answer
33 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
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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!
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0answers
38 views

Michael Artin Homework. [on hold]

Prove that a rigid motion is bijective. | m(X)-m(Y) | = |X-Y| how can i prove this ? I have no idea how to start . Its from Michael Artin book .
2
votes
2answers
38 views

Size of the orbits of a normal subgroup

So this is the question: Let $H$ be a finite subgroup of $G$, and let $(h,h')(x)=hxh^{-1}$ define an achtion of $H\times H$ on $G$, prove that $H$ is a normal subgroup of $G$ if and only if every ...
2
votes
1answer
45 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 ...
0
votes
1answer
36 views

if $A$ and $B$ are subnormal, then $A\cap B$ is subnormal [on hold]

A subgroup $X$ of a group $G$ is said to be subnormal if there exists a series $$X=X_0\subseteq X_1\subseteq\cdots\subseteq X_n=G$$ where each $X_i$ is normal in $X_{i+1}$. Prove that if $A$ and $B$ ...
0
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1answer
35 views

Finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ [on hold]

Give an example of a finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ of $G$ This question is from Herstein Topics in ...
3
votes
0answers
48 views

Subgroups of $S_{n}$ of index $n$

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups ...
2
votes
2answers
35 views

Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)

I am a beginner in group theory and I'm looking for finite groups that satisfy some properties. The only example I've found so far is: $$G_{q,c} = \{ f: \mathbb{F}_{q} \to \mathbb{F}_{q}, z \mapsto ...
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0answers
12 views

Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
0
votes
1answer
18 views

$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Let $G$ be a p-group. $|G|=p^n$ for some n. Let X be a finite set so that $\,p\nmid |X|\,$, G acts upon X. Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$ I am trying to show ...
20
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2answers
77 views

any $2$-dimensional rep of a finite, non-abelian simple group is trivial

Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering ...
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3answers
105 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 ...
1
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1answer
39 views

Group objects in the category of rings

Are there group objects in: $\text{Ring}$ $\text{CRing}$ If so, why doesn't anyone talk about them? On the other hand, $$ \begin{align} cogroup \ objects \ in \ \text{CRing} &= co(group \ ...
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1answer
35 views

List all the elements of the subgroup of Möbius transformations preserving the set $\{0, 1 + i, \infty\}$

List all the elements of the the subgroup $M_{\{0, 1, \infty\}}$ of the group of Möbius transformations, preserving the set $\{0, 1, \infty\}$ and give an explicit isomorphism $M_{\{0, 1, \infty\}} = ...
1
vote
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 ...
1
vote
3answers
32 views

Intersect of Stabilizers is a Normal Group

I am looking for guidance for two problems on group action, one of them is here and the other one will be posted in another page: Assume that $G$ operates on a set $\Omega.$ Show that ...
3
votes
1answer
37 views

Proving a relation for representations of gauge groups

Let ${\cal G}$ be a Lie group - possibly disconnected. Let ${\mathfrak g}$ denote the corresponding Lie algebra. Let $R_k$ be a general unitary representation of ${\cal G}$ and $R$ be the adjoint ...
2
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0answers
57 views

Which convex $2n$-gons have symmetry group $D_n$ instead of $D_{2n}$?

The equilateral octagon $M$ in the first image has the same symmetry group as the small embedded square - namely the dihedral group $D_4$ - with $8$ elements and generators ${x,y}$ with $x^4 = e, y^2 ...
0
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0answers
16 views

Lattice of subgroups

I'm trying to find the lattice of subgroups of the symmetric group $\mathfrak S_3$ and of the diedral group $\mathcal D_8$ (the group of order 8). I searched on google, but I didn't find anything.
0
votes
0answers
30 views

On subgroups of the form $HZ(G)$ where $H$ is abelian subgroup of non-abelian group $G$ such that $H \rlap{\;\,/}\subseteq Z(G)$

Let $H$ be a an abelian subgroup of a non-abelian group $G$ such that $H \rlap{\;\,/}\subseteq Z(G) Z(G)$ ; then I can prove that $HZ(G)$ is an abelian subgroup such that $Z(G) \subset HZ(G) \subset ...
1
vote
2answers
29 views

Is there any automorphism $f$ that satisfies these requirements? [on hold]

Suppose that $\Bbb R $ is the set of real numbers. Is there any automorphism $f$ from $(\mathbb R,+)$ to $(\mathbb R,+)$ of the following form? $$f(x)=kx, \quad k \neq 0,1 , \quad k \in \Bbb R$$