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

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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
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1answer
17 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
6 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 ...
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1answer
14 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 ...
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0answers
22 views

Show that $(ℝ,*)$ is a group with $\phi(1)$ as the identity element [on hold]

Let $\phi:{\Bbb R}\to{\Bbb R}$ be any bijection and define the binary operation $$x*y=\phi\bigl(\phi^{-1}(x)\phi^{-1}(y)\bigr)\ .$$ Then show that $(ℝ,*)$ is a group with $\phi(1)$ as the identity ...
20
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2answers
67 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
60 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
33 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
17 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\}} = ...
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2answers
34 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
28 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 ...
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0answers
24 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 ...
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4answers
125 views

how to show $\mathbf{Q} $ is not free

we know torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show Q is not finitely generated? and not free?
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0answers
52 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 ...
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0answers
14 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.
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28 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 ...
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2answers
28 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? $$f(x)=kx, \quad k \neq 0,1 , \quad k \in \Bbb R$$
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1answer
106 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
49 views

Number of group actions [on hold]

In how many ways can the group $\mathbb{Z}_5$ act on $\{1,2,3,4,5\}$.
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0answers
28 views

Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
2
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2answers
41 views

Show that $f _ a $ is a Homomorphism

For a fixed element $a$ is a group $G$, define $$f _ a (x) = a ^ {−1} xa , x \in G$$ Show that $f _ a $ is a Homomorphism. I know that to show that a mapping $f:G \rightarrow G'$, Where $G$ and $G'$ ...
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2answers
49 views

How to find number of subgroups of $\mathbb Z_8+\mathbb Z_2$ [on hold]

How to find number of subgroups of order $8$ in $\mathbb Z_8+\mathbb Z_2$? Any hint is welcome
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2answers
69 views

Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
1
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1answer
23 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 ...
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1answer
84 views

Is $(G,*)$ commutative? [on hold]

$(G,*)$ is a group and for some three consecutive integers $i=j,j+1,j+2$, it satisfies $(a*b)^i=a^i*b^i$ for every $a,b\in G$. Is $(G,*)$ commutative?
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1answer
25 views

Are the principal congruence subgroups of SL(2,Z) normally generated by a single element?

Let $N\ge 3$, then would I be correct in saying that the principal congruence subgroup $\Gamma(N)$ (defined to be the 2x2 matrices in $SL(2,\mathbb{Z}$) congruent to the identity mod $N$) is the ...
5
votes
2answers
59 views

Normal subgroups in groups of odd order

I put the following question in my first-year algebra final this year: Suppose $G$ is a finite group of odd order and $N$ is a normal subgroup of order $5$. Show that $N\le Z(G)$. (By the way, this ...
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26 views

Isomorphism type of a finite matrix group

Give a known group to which $SO_5$ is isomorphic. Where $SO_5$ consists of $$\begin{pmatrix} \cos\frac{2k\pi}{5} & \sin\frac{2k\pi}{5} \\ -\sin\frac{2k\pi}{5} & \cos\frac{2k\pi}{5} ...
1
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1answer
47 views

Group actions: Why do we place the condition that $S$ be finite in the following theorem?

Theorem. Let $G$ be a group, $S$ be a $G$-set, and $S$ be finite, then $$|S|= \sum_{a \in A} [G : G_a],$$ where $A$ is a subset of $S$containing exactly one element from each orbit $[a]$. Here, $G_a$ ...
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21 views

Please show that $W$ is a normal subspace of $\sigma$. [on hold]

Suppose that $\sigma$ is a symmetry transformation, $V$ is a space, and $W$ is a subspace of $V$. Please show that $W$ is a normal subspace of $\sigma$.
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1answer
26 views

minimal number of relations in nonabelian dihedral groups of order a power of 2

In my recent minimal number of relations r question for the symmetric group of degree 3, it was demonstrated that r = 2. What is r in general for nonabelian dihedral groups of order a power of 2? ...
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2answers
57 views

To show that $\langle\mathbb Q,+\rangle$ is not isomorphic to $\langle\mathbb Q \setminus\{0\} , \,\cdot\,\rangle$

For this question is it enough to say that they both dont have same cardinality so they are not isomorphic.Can we exhibhit any structural property diference here? THANKS
1
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1answer
31 views

Show that $H$ is transitive on the set $G$.

Let $G$ be a group and let a be a fixed element of $G$. The map $\lambda_{a}: G \to G$, given by $\lambda_{a}(g) = ag$ for all $g \in G$, is a permutation of the set $G$. Note $H = \{\lambda_{a} ...
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1answer
37 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
82 views

Is my understanding for group algebra correct?

Let $G$ be a group, $k$ be a commutative unital ring. Consider $\mathbf{Alg}_k^{inv}$ the category of unital $k$-algebras whose multiplicative semigroup is a group. Then there is a forgetful functor ...
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1answer
42 views

Representing group by permutaions

Let the group is given by this relations $<a,b\ |\ a^5=b^4=e,bab^{-1}=a^2>$. I am asked to find the cycle index of this group. In order to find cycle index I need to represent the group with ...
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0answers
50 views

A name for the automorphisms induced by the normalizer by conjugation?

Let $N$ be a subgroup of a group $G$, $H := \operatorname{N}\left( N \right)$ the normalizer of $N$. There is a natural morphism from $H$ to $\operatorname{Aut}\left( N \right)$ given by $h \mapsto ...
3
votes
1answer
46 views

Constructing an irreducible representation for a finite group

This is not a homework. Recently, I have been starting to study the representation and character theory and I am doing some exercises in this field, but I have some problem with some of them, like: ...
2
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1answer
58 views

Show that a simple group of order 60 has no proper subgroup of order greater than 12

I am trying to show that any simple group of order $60$ has no proper subgroup of order greater than $12$. I know that $G$ is isomorphic to $A_5$, a non-abelian simple group of order $60$. I suppose ...
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2answers
21 views

Statement of Sylow's Fourth Theorem (single conjugacy class)

I am confused by the statement of Sylow's Fourth Theorem: Let $G$ be a finite group, $p$ a prime. The Sylow $p$-subgroups of $G$ form a single conjugacy class of subgroups. In particular, I do not ...
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2answers
37 views

Alternative proof of '$I$ is maximal iff $R/I$ is a field'

For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field. I write my own proof and it checks with the 'traditional' proof which ...
2
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2answers
64 views

What can we say about the order of a group?

Let $G$ be a group and $a ∈ G$. If $a^{12}= e$, what can we say about the order of $a$? What can we say about the order of $G$? We know that $|a|$ divides $12$, but what can we say about the order of ...
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0answers
59 views

Groups with more than half elements of order 2

In Dihedral groups, at least half elements are of order two. Question: If a (non-abelian) finite group has at least half elements order two, then what can be said about the group?
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2answers
43 views

$K \unlhd H\unlhd G$, K is sylow in H, proof $K \unlhd G$

$K \unlhd H\unlhd G$, K is sylow in H, proof $K \unlhd G$ I know in general it's false, I wonder how should I use the condition that K is sylow? is it that the K is the unique sylow subgroup? and ...
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1answer
40 views

Quotient Objects in $\mathsf{Grp}$ II

This question is a sort of continuation of a previous one. In CWM, Maclane says ... every quotient object of a group $G$ in $\mathsf{Grp}$ is represented by the projection $\pi:G\rightarrow G/N$ ...
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3answers
50 views

Mapping Normalizer to Automorphism

I am squeezing my brain trying to understand this problem: Let H be a subgroup of $G$, and denoted by $\operatorname{Aut}(H)$ the group of all automorphisms of $H. $ Define $$\alpha : N_G(H) ...
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0answers
31 views

If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
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2answers
53 views

A question in finite group theory with proof using representation theory

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises in this field, but I have a problem with an exercise: I want to prove ...
0
votes
1answer
33 views

commuting subsets in a group

For a countable infinite discrete group $G$, consider the following three properties. (P1) $G$ is abelian. (P2) For any finite subset $K$ of $G$, there exists an element $s\in G$, such that the two ...
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2answers
37 views

order of group generated by two element with some relation.

The group defined by generators $a,b$ and relations $a^{8}=b^{2}a^{4}=ab^{-1}ab=e$ has order at most 16. How to prove that? I have no idea.