0
votes
3answers
35 views

Calculate quotients of $\mathbb S_3$ and $\mathbb D_4$

Problem Calculate all the quotients by normal subgroups of $\mathbb S_3$ and $\mathbb D_4$,i.e., charactertize all the groups that can be obtained as quotients of the mentioned groups. For the case ...
1
vote
1answer
17 views

Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
0
votes
2answers
49 views

Proving certain groups are normal

Given the following subgroup of $\mathbb S_4$: $$K=\{\mathrm{id},(12)(34),(13)(24),(14)(23)\}$$ prove that $K \unlhd \mathbb A_4$, $K \unlhd \mathbb S_4$ I am trying to solve this problem doing as ...
2
votes
1answer
24 views

left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
0
votes
2answers
44 views

Number of non isomorphic groups of order $122$, My attempt through Sylow theory.

$|G|=122 = 2 . 61$ No. of sylow $2$ subgroups $= 1$ or $61 = n_2$ No. of sylow $61$ subgroups $= 1 = n_{61}$ Let the group of order $61$ be $H_{61}$ and the group of order $2$ be $H_2$ Then : ...
9
votes
1answer
107 views

Cardinality of $\text{Aut}(G\times G) $

Let $G$ be a finite group. If $|\text{Aut}(G)|$ is known, what can we say about $|\text{Aut}(G\times G)|$ ?
0
votes
0answers
12 views

Compounding unary operators

I am working with the symmetric group $S_5$. I have 3 unary operators defined: $R$, $T$, and $O$, and I'm writing about their composition. Suppose I want to denote the compound operation of "$T$, ...
2
votes
2answers
25 views

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$ Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} ...
0
votes
2answers
36 views

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$. Attempt: $|G|=168=2^3.3.7$ Then number of sylow $7$ subgroups in $G ...
3
votes
0answers
32 views

If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
2
votes
0answers
54 views

Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups. Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups. Definition: ...
1
vote
2answers
67 views

Does $G$ always have a subgroup isomorphic to $G/N$?

Let $G$ be a group and $N$ a normal subgroup of $G$. Must $G$ contain a subgroup isomorphic to $G/N$? My first guess is no, but by the fundamental theorem of abelian groups it is true for finite ...
2
votes
1answer
40 views

Smallest possible odd integer that can be the order of a non-Abelian group.

Smallest possible odd integer that can be the order of a non-Abelian group. Attempt: A non abelian group means $Z(G) \subset G$ . Hence, it suffices to find the smallest odd integer $n$ such that ...
2
votes
1answer
48 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
1
vote
1answer
106 views

$A\rtimes B$ vs $B\rtimes A$?

What is the difference between $A\rtimes B$ and $B\rtimes A$? Could one group be normal in a group $G$ and at the same time not normal in another group? The asymmetry of the $\rtimes$ symbol (with ...
1
vote
1answer
38 views

For every element of a finite group, there are two distinct exponents that produce the same power

I'm starting my group theory course and we arrived to the following demonstration: Let $G$ be a finite group, then, for every $a\in G$, $O(a) \leq |G|$ where $O(a)$ is the order of the element $a$ ...
0
votes
2answers
57 views

Semidirect Product variations

Is there two semidirect products of order $16$ with $C_2$ normal subgroup? How many groups of the form $C_4 \times C_2$ : $C_2$ are there ? Is it one expression for two groups ? or more?
7
votes
2answers
105 views

Uniqueness of the direct product decomposition of finite groups

A group $G$ is indecomposable if: $G = H \times K \Rightarrow \{ H,K \} = \{1, G \}$. Then, a finite group $G$ decomposes into a direct product of indecomposable groups: $G = \prod_i G_i$. ...
2
votes
1answer
42 views

The only group of order $255$ is $\mathbb Z_{255}$ ( Using Sylow and the $N/C$ Theorem)

Let $G$ be a group such that $G =255 = 3.5.17$. Let $H$ be a sylow $17$ sub group of $G$. Then, by Sylow's theorem : the number of sylow $17$ sub groups can be $n=1,18,35,...$ . Since, $n$ should ...
4
votes
1answer
76 views

Automorphism $f$ so that $f(x)=x^{-1}$ for half the members of the group: is it an involution?

Let $G$ a finite group. Let $f: G \to G$ an automorphism such that at least half the elements of the group are sent to their inverses, i.e $$\mathrm{card}(\{g \in G|f(g) = g^{-1}\}) \geq ...
1
vote
1answer
30 views

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$ where $cl ~(a)$ refers to the conjugacy class of $a$. The proof in the book which I am reading asks to prove ...
1
vote
0answers
40 views

Relation between $|H \lor K|$ , $|H|$ and $|K|$

Let $H$ and $K$ be subgroups of a finite group , then we know that the subgroup generated by $H \cup K$ i.e. $H \lor K$ is the smallest subgroup containing both $H$ and $K$ , then how can we relate ...
1
vote
1answer
72 views

Are any two groups of order 23 isomorphic to each other?

I have to decide whether the following statements are true or false, with proofs. Any two abelian groups of order $23$ are isomorphic to each other Any two abelian groups of order $25$ are ...
3
votes
1answer
35 views

Defining a subgroup of $GL(2,7)$ in GAP

Considering this resent post in which $|G|=42$, I am thinking of making this subgroup concrete in GAP environment. Maybe, if the structure of $G$ was known then, we would use an appropriate mapping ...
1
vote
3answers
50 views

Subgroup of matrices exercise

Let $G=\{\left( \begin{array}{ccc}1 & b \\ 0 & a \\ \end{array} \right) : a,b \in \mathbb Z_7, a \neq 0\}$. Find the order of $G$. For each prime $p$ such that $p$ divides $|G|$, find all ...
9
votes
2answers
59 views

order of an elliptic curve

I have found that the curve given by $x^3+x+1=y^2$ over $\mathbb{F_5}$ has 9 points. Now I am supposed to find the number of points of the same curve on $\mathbb{F}_{125}$. Using Hasse and the fact ...
3
votes
3answers
62 views

Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
5
votes
2answers
71 views

If $p\mid|G|$ then how many elements of order $p$ are there in $G$?

Let $G$ be a finite group and $p$ be a prime such that $p\mid|G|$ , then obviously $G$ has an element of order $p$ (by Cauchy's theorem) ; I would like to know exactly how many elements of order $p$ ...
0
votes
1answer
80 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
1
vote
6answers
66 views

Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
5
votes
0answers
50 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
1
vote
1answer
73 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
2
votes
1answer
64 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
1
vote
0answers
56 views

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $$|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$$ for each b fixed & for all $a+b$ such that ...
1
vote
2answers
24 views

Does the following binary operation form a group on a set with 3 elements? (multiple identities?)

Let S = {a, b, c}. *| a b c ----------- a| a b c b| b a a c| c a a This seems to have all the desired characteristics of a group, however, both b and c ...
4
votes
1answer
45 views

How to determine the isomorphism types of given groups with generators and relations

I was classifying the all groups of order 30 and I got the following groups $\langle a,b \mid b^{-1}ab=a^4, a^{15}=b^{2}=1\rangle$ and $\langle a,b \mid b^{-1}ab=a^{11}, a^{15}=b^{2}=1\rangle$. How ...
1
vote
0answers
23 views

How to reconstruct geometric object that a Frobenius group acts on

A Frobenius group has equivalent definitions: a transitive permutation group on a finite set such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. ...
2
votes
1answer
53 views

$GL_2(R)$ and $\mathbb{A}_4$

So, i have to prove that $\mathbb{A}_4$ is not isomorphic to a subgroup $G$ of $GL_2(R)$. Here is a "hint" (are previuos part of the same problem that i was thinking, so i supose should help): If ...
0
votes
2answers
30 views

Show that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+) $ holds

I want to show, that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+)$ holds with $p$ prime. ($\mathbb{F}_p^+$ is the additive group, $\mathbb{F}_p^\times$ multiplicative group) As hint we ...
1
vote
1answer
33 views

Proof of the theorem that relates the size of the conjugacy class to the order of the centralizer subgroup

Could you please explain the step in this theorem and proof that I do not follow: If $G$ is a finite group, let $C_{G}(x)$ be the centralizer of $x$ in $G$, that is $C_{G}(x)=\{g \in G: xg=gx\}$ ...
2
votes
3answers
64 views

Motivation and intuition of double cosets

In Dummit and Foote, there was a problem on double cosets. Let $H$ and $K$ be subgroups of $G$ and $x\in G$, $HxK$ is the double coset of $x$. I can prove the double cosets partitions $G$. However, I ...
0
votes
2answers
64 views

Use Sylow's theorem to show that $G = HN_G(P)$

I am stumped on this question. Does anyone have some helpful hints or a solution to this question? Thanks! Let $G$ be a group of finite order. Let $H$ be a normal subgroup of $G.$ Let $P$ be a ...
4
votes
2answers
110 views

Normalizer and centralizer of abelian subgroups of a group are equal [closed]

I have a question: Let $G$ be a finite group. If for each abelian subgroup $H$ of $G$ the centralizer and the normalizer of $H$ are equal, that is, $C_G(H)=N_G(H)$, prove that $G$ is abelian ...
10
votes
3answers
88 views

Group of order $224$

Any one can give a hint to prove this? Every group of order $224$ have a subgroup of order $28$.
5
votes
2answers
58 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
0
votes
0answers
67 views

Let $G$ be a group of order 30. Show that $G$ has a subgroup of order 15.

Let $G$ be a group of order 30. Show that $G$ has a subgroup of order 15. Now it has a Sylow 3 & a Sylow 5 subgroup. Next what?
1
vote
1answer
25 views

Frobenius dihedral groups

How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?
2
votes
1answer
52 views

Characteristically simple subgroup

Let $G$ be a finite group and $H$ be a minimal normal subgroup of $G$. How can we show that $H$ is a characteristically simple subgroup of $G$?
0
votes
0answers
23 views

finite simple group of Lie type

let $G$ be a finite simple group of Lie type. would someone please help me! I want to know, is the following expression true or not? " There exists a simple linear algebraic $\mathcal{G}$ over an ...
2
votes
1answer
37 views

List of groups with specific divisors

I want to find list of finite simple nonabelian groups which their orders divisor lies in specific set of primes for example the set $\{2,3,5,7,11\}$. Is there a method to do this? Would Zsigmondy's ...