Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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5
votes
2answers
180 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
1
vote
3answers
80 views

Need help determining what this center is isomorphic to

I am looking at this released exam, problem 2c. It states: Let $G$ be a non-abelian group of order $8$ and $Z$ be the center of $G$. To which group is $Z$ isomorphic? It gives a hint to recall a ...
0
votes
0answers
50 views

A finite group which is isomorphic to $PSL(2,p)$

Let $p$, $q$ be a odd prime numbers such that $p=2q+1$. Let G be a finite group of order $\frac{p(p^2-1)}{2}$. If all Sylow 2-, p-, and q-subgroups are not normal, G is isomorphic to $PSL(2,p)$. The ...
2
votes
1answer
74 views

Is O(1) a Lie Group?

In reading Georgi (Lie algebra in particle physics) I reaf at page 43 the following definition of Lie Gruoup: "a lie gruoup is a group whose elements depend smoothly on a set of continuous ...
1
vote
2answers
49 views

If $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$

Question 1. Let $K\subseteq H\subseteq G$ and if $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$. 2. Let $K\subseteq H\subseteq G$ and if $K$ is a subgroup of $G$ ...
1
vote
1answer
41 views

Existence of non-abelian split metacyclic extensions.

Is there any necessary conditions that must be held in order to guarantee the existence of a non-abelian split metacyclic extension? i.e. for which $m,n\in\mathbb{Z}$ there exist a non abelian split ...
2
votes
2answers
111 views

Direct product and Sylow subgroups

Let $G$ be a finite group that is equal to inner direct product of its subgroup $P$ and $Q$, where $P$ is a Sylow $p$-subgroup and $Q$ is a Sylow $q$-subgroup of $G$. If $L \le G$, prove that $L$ is ...
0
votes
1answer
54 views

Difficulty with a lemma needed to prove $A_n$ is a simple group for $n>4$

The theorem is: For $n \geq 5$, every normal subgroup $N$ of $A_n$ contains a $3$- cycle. The proof starts like this: Let $\sigma$ be an arbitrary element in a normal subgroup $N$. There are ...
1
vote
2answers
106 views

On a classification of all the characteristic subgroups of a finite abelian $p$-group.

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is ...
1
vote
1answer
69 views

Is a finite group which is generated by two characteristic abelian subgroup always abelian?

Let $G$ be a finite group. If there exist two characteristic subgroups $H,K$ of $G$ such that $H$ and $K$ are abelian and generate the whole group $G$. Then can we conclude that $G$ is abelian? All ...
4
votes
2answers
118 views

Group of order $135$ abelian and not cyclic

I am trying to solve the following: Let $G$ be a group of order $135$. Show that if $G$ has more than one normal subgroup of order $3$, then $G$ is abelian and non-cyclic. What I could do was: ...
0
votes
1answer
153 views

Finite abelian groups of order 100

(a) What are the finite abelian groups of order 100 up to isomorphism? (b) Say $G$ is a finite abelian group of order 100 which contains an element of order 20 and no element with larger order. Then ...
1
vote
3answers
98 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
1
vote
1answer
51 views

Basic doubt about cosets

Studying some basic group theory I had the following doubt: For $H$ subgroup of a finite group $G$ (doesn't matter invariance of $H$), is it true that $$|G/H|=|\{aHa^{-1}:a \in G\}| \space ...
0
votes
2answers
31 views

If $k|n, k \geq 2$, then $D_{n}$ has a subgroup isomorphic to $D_{k}$

Restatement of question: If $k|n, k \geq 2$, then the group $D_{n}$ has a subgroup isomorphic to the group $D_{k}$. My attempt at proving the result stated: Let us say that $D_{n}= \{1, \sigma, ...
1
vote
1answer
78 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
1
vote
0answers
38 views

Given a space group, how to determine which layer groups are its subgroups?

I am studying various crystals and the two-dimensional materials that could be potentially obtained by cleaving them (isolating a region bounded by two parallel planes). In elucidating the properties ...
2
votes
0answers
32 views

How to find the power of generator defined over finite field , $\mathbb F_{2 ^m}$?

List item Actually ,I am trying to execute the algorithm to find the power of generator of field(group) as shown in table of attached file but when k=7 and onward ,I could not understand what is ...
0
votes
3answers
80 views

Group Theory: How do I determine if an element generates a group?

I was asked if the group $(Z_{17} \setminus \{0\}, \cdot)$ is generated by the element $2$. I understand the concept of generating sub-groups in group theory. If I was given a group $G$ and asked to ...
2
votes
4answers
556 views

Why do Z/7 have no cubic root of 2?

I was reading a textbook and came across the following line: Now we prove there is no cube root of 2 in $Z/7$. By noting that $(Z/7)^\times$ is cyclic of order 6, it will have only two third ...
1
vote
2answers
80 views

$|G|=p_1p_2p_3$ distinct primes with $p_i \nmid p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \nmid p_j-1$ for all $1\leq i,j \leq 3$ and let $G$ be a group of order $p_1p_2p_3$. Show that $G$ is cyclic. I've tried to come up with ...
0
votes
1answer
109 views

Real regular representation of cyclic group

I am looking for help to answer the following questions: What are the irreducible real representions $ρ: C_n → GL(V ) $ of a cyclic group of order n? How does the real regular representation $RC_n$ ...
3
votes
1answer
73 views

Isomorphism of two non-abelian groups of order $pq$

Let $p$ and $q$ be two primes such that $q\mid p-1$. Suppose $\phi, \varphi$ are two non-trivial homomorphism from $\mathbb{Z}_q$ to $Aut(\mathbb{Z}_p)$. How to define an isomorphism from ...
0
votes
2answers
68 views

Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
2
votes
2answers
68 views

Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
2
votes
1answer
65 views

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
2
votes
4answers
240 views

If G is a group not cyclic then its order can be:

If G is a group not cyclic then its order can be: a)15 b)35 c)77 d)120 e)2011 Well, i know that if G is not cyclic then it is not isomorphic to Zn, but i think it does not help much. Any ...
-1
votes
2answers
126 views

Equivalent permutation representations.

The definition of Equivalent Permutation Representations that is defined in "A course in Theory of Groups" by Derek Robinson Suppose we have group $G$ has permutation representation on set $X$ and ...
0
votes
2answers
77 views

Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G:G]$.

Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G:G]$. I have already figured out that $|Z(G)| = p$, and that $G'=[G,G] \lhd G$. Also, $|G'| = p$ or ...
-2
votes
1answer
50 views

What are all the subgroups of $\Bbb{Z}/10\Bbb{Z}$?

I thought they would be $\Bbb{Z}/n\Bbb{Z}$, where $1 \leq n \leq 10 $. What is wrong in that ? And, how is $\Bbb{Z}/n\Bbb{Z}$ a cyclic group. Is it's generator the element 1 ?
0
votes
0answers
19 views

scalar multiple of Young symmetriser

The following is a lemma on Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
1
vote
1answer
59 views

Normal subgroups and index problem

Let $G$ be a finite group and $H$ and $K$ subgroups such that $H \lhd G$ or $K \lhd G$. If $gcd(|K|;|G:H|)=1$, show that $K \subset H$. I think I could prove it for the case $H \lhd G$ Let $k \in ...
0
votes
1answer
66 views

How does GAP understand $SL_2(\mathbb{F}_3)$?

When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats the field on which the representations are. There GAP ...
1
vote
1answer
74 views

Prove the existence of order 4 subgroups of order 8 groups

I am participating in an Introductory course in groups and I have the following question: Let $G$ be a finite group of order $8$. Prove that $G$ has a subgroup of order $4$ and a subgroup of order ...
3
votes
2answers
68 views

Finite subgroup of $\mathbb C^{\times}$

I was trying to show that every finite subgroup of $\mathbb C^{\times}$ is equal to $G_n$ (the nth roots of unity) for some $n \in \mathbb N$ without invoking Lagrange's theorem, I got stuck at one ...
0
votes
1answer
33 views

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p = q$.

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p=q$. First of all, I'm a bit confused about the meaning of '... are conjugate'. Does this mean that ...
0
votes
1answer
64 views

Finite matrix groups as subgroups of $S_n$.

I have heard that all finite subgroups are isomorphic to a subgroup of $S_n$. I was thinking about examples of this. In particular I would like to know how this works for certain matrix groups. The ...
2
votes
1answer
51 views

Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
2
votes
2answers
50 views

What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?

It seems that there is a action by which $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ permute the $p^2$ ordered tuples in $\mathbb{F}_p^2$. What is the map from the $2 \times 2$ matrices over ...
3
votes
2answers
128 views

How did we classify the finite simple groups when we haven't classified the primes?

Why was the classification of the finite non-abelian simple groups "easier" (!!) than the classification of the finite abelian simple groups [the prime numbers], which still doesn't exist? (Does it? ...
0
votes
1answer
52 views

Prove that any group is a disjoint union of conjugacy classes.

How do I prove that any group is a disjoint union of conjugacy classes? Any reading reference would also be helpful
0
votes
1answer
100 views

How to read GAP's output on “IrreducibleRepresentations”?

For example for the group $SL_2(\mathbb{F}_3)$ I get the following, ...
1
vote
1answer
21 views

Finitely presentated subgroups of a group are normal?

If a group is finitely generated, then it is a quotient of the free group on the set of generators. Further if a subgroup of a group has some finite presentation, does it mean that it is normal, ...
7
votes
0answers
196 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and ...
0
votes
1answer
45 views

Are these groups of order $81$ isomorphic?

The classification of groups of order $p^4$ is well known. However, different sources sometimes classify in different way. I am trying to compare, in the case $p=3$ between Burnside list (p. 100-102) ...
1
vote
1answer
64 views

Lifting representations, kernels and invariant subspaces

Let $G$ be a group, $N \triangleleft G$, $G/N$ the corresponding quotient group. Suppose $\rho : G/N \longrightarrow GL(\mathbb{C})$ is a representation of $G/N$. Then the composition ...
0
votes
2answers
65 views

Action of $G$ on the left cosets of $H$ giving a non-trivial homomorphism

If $H < A_5$ is a subgroup of index $3$, the action of $G$ on the left cosets of $H$ gives a non-trivial homomorphism $$\underset{order \ 60}G \rightarrow \underset{order \ 6}{S_3}$$ which ...
0
votes
1answer
56 views

Considering transitive $G$-set

Question. Suppose that $X$ is a transitive $G$-set of size greater than $1$ and let $\pi$ be the associated permutation representation with the character $\chi$. Show that some element $g \in G$ ...
3
votes
2answers
77 views

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
0
votes
0answers
31 views

Inducing $A_4$ from $\langle (123) \rangle$

Let $G=A_4$ and $H=\langle (123) \rangle < G$. Compute $Ind_{H}^G \chi$ for every irreducible $\chi$ of $H$. Choose the right transversal of $H$ in $G$ as $V_4=\{ 1, (12)(34),(13)(24),(14)(23) ...