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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3
votes
1answer
88 views

Action of $S_7$ on the set of $3$-subsets of $\Omega$

Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem: Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
5
votes
0answers
466 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
3
votes
2answers
310 views

Finite group is generated by a set of representatives of conjugacy classes.

Could you tell me how to prove that a finite group is generated by a set of representatives of conjugacy classes? I've read this ...
1
vote
2answers
910 views

Show groups of symmetries of a cube and a tetrahedron are not conjugate in isometry group.

I've shown that the symmetry group of a cube and a tetrahedron are both isomorphic to S4, but I am now trying to show that they are not conjugate when considered as subgroups of isometries of 3D ...
4
votes
1answer
118 views

Group homomorphisms into a field

Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
0
votes
2answers
134 views

Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$

Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$. Here is what I have here: Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
6
votes
3answers
643 views

Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.

Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements. We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$. If $G$ is ...
3
votes
1answer
106 views

Infinite products of a (finite) group

So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
4
votes
2answers
125 views

Finite abelian $2$-group

If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors). Any ...
8
votes
2answers
45 views

The index of $\xi_4^*$ in $\xi_4$

Just seeing if i'm right: With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative ...
4
votes
1answer
123 views

sylow basis of finite solvable groups

Let ‎‎$G‎‎$‎ be a‎ ‎finite ‎solvable non-$p$-group ‎and ‎‎$‎‎A$ ‎be a ‎‎‎‎maximal ‎subgroup ‎of ‎‎$G‎‎$‎. ‎ Therefore ‎$A‎‎$ ‎is ‎of ‎primary ‎index ‎‎$p^{n}‎ $‎‎, that is ‎$|G : A|=p^{n}‎$ ‎‎where ...
3
votes
1answer
49 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
1
vote
1answer
436 views

Non trivial Automorphism [duplicate]

Prove that every finite group having more than two elements has a nontrivial Automorphism. It is from Topics in Algebra by Herstein. I am not able to solve.
5
votes
3answers
176 views

Finding the subgroup of $(\mathbb Z_{56},+)$ which is isomorphic with $(\mathbb Z_{14},+)$ by GAP

I am sorting some easy questions for the students in Group Theory I. One of them is: Is $(\mathbb Z_{14},+)$ isomorphic to a subgroup of $(\mathbb Z_{35},+)$? What about $(\mathbb Z_{56},+)$? I ...
7
votes
3answers
167 views

Homomorphism from $\mathbb{Z}/n\mathbb{Z}$

Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
6
votes
2answers
81 views

Embeddings of $GL(n-1,q)$ into $GL(n,q)$

Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
3
votes
2answers
148 views

Rotman's Introduction to to the theory of groups. Exercise 3.45.

Can you give me a hint on the first part of the exercise? Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
8
votes
1answer
105 views

Cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$?

What are the cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$, the general linear group over the finite field of order $p$, where $p$ is prime? Obviously, each cyclic subgroup is generated by some ...
0
votes
1answer
240 views

Classify the abelian groups of order 81, 144 and 216

Classify the abelian groups of order n, 2n, 4n
2
votes
2answers
166 views

$H$ must contain every Sylow $p$-subgroup of $G$

Let $p$ be a prime factor of the order of a finite group $G$. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that $H$ must contain every Sylow $p$-subgroup of $G$.
4
votes
2answers
336 views

If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.

If the order of a finite abelian group is square free, show that the group is cyclic. This is a question from "basic abstract algebra" by bhattacharya
7
votes
1answer
1k views

Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$.

Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$. Let me start off with what I did: Assume $G$ is abelian. Then we know ...
2
votes
2answers
116 views

Sum of two squares in a $\Bbb Z/p\Bbb Z$

I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of ...
1
vote
1answer
183 views

$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?

In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
1
vote
0answers
99 views

Generators in $p$-groups

Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators ...
8
votes
1answer
146 views

Why does the automorphism used to construct the group have to be non-inner?

I have a question on why a particular assumption is made that the automorphism used to construct a certain group be non-inner. In [Herstein, Topics in Algebra, p. 69], a construction of a nonabelian ...
1
vote
1answer
82 views

An question of $\pi$-groups for the alternating group $A_5$

Let $G=A_5$, the alternating group of degree 5. Please help me show that for $\pi = \{2,3\}$ we have that $M \leq G$ is a maximal $\pi$-group if and only if $M \cong A_4$ or $M \cong S_3$ (the ...
1
vote
2answers
304 views

Multiplicative group of integers modulo n

Consider the abelian group $U_n=\{a\in \mathbb{Z}_n:(a,n)=1\}$. Is there a natural way to understand it as a subgroup of any other interpretation of the cyclic group of order $n$. For example, ...
8
votes
1answer
252 views

How to recover the integral group ring?

I would like to solve the following exercise: Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
3
votes
0answers
51 views

Any finite subgroup of the abelian group $(F-\{0\},\cdot)$ is cyclic? ($F$ a field) [duplicate]

I found this problem: Suppose that $F$ is a field, and that $(F-\{0\},\cdot)$ is an abelian group. Show that if $H$ is a finite subgroup of $F-\{0\}$, then $H$ is cyclic. What I have done is: ...
2
votes
1answer
64 views

About commutators and center o a certain group

Let $G$ be a group and $H, K\unlhd\ G$ (normal subgroups), with $K\leq H$ such that $\left[H, Aut(G)\right]\leq K$. Why does this imply that $H/K\leq Z(G/K)$? $Z(G)$ denotes the center of $G$.
2
votes
1answer
115 views

How many elements of order $7$ are there in a group of order $28$ without Sylow's theorem

How many elements of order $7$ are there in a group of order $28$ I need to prove this result without using the Sylow's Theorem.By Sylow's Theorem it has only one subgroup and the anser becomes ...
10
votes
1answer
233 views

What can we say about the size of $HK\cap KH$ when $HK\neq KH$?

If $G$ is a finite group, and $H$, $K$ are proper subgroups of $G$, then it is not necessary that $HK=KH$. But, these two subsets have same size. The question I would like to ask, then, is If ...
2
votes
4answers
86 views

What is the meaning of the following quotient group?

$S_4 / \{(1 \, 2)(3 \, 4), (1), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \}$ We let $H=\{(1 \, 2)(3 \, 4), (1), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \}$ be a subgroup. Then the above notation is the set of ...
3
votes
1answer
56 views

Sharply t-transitive groups.

Suppose that $G$ acts sharply $t$-transitively on a set $\{1,\cdots, n\}$. Then I want to show that if $n = t + 2$ then $G = A_n$. I can indeed show this, but I feel it's unnecessarily messy. My way ...
16
votes
2answers
385 views

Which finite groups are the group of units of some ring?

Motivated by this question: Which finite groups are the group of units of some ring? Which finite groups are the group of units of some finite ring? Which finite abelian groups are the group of ...
7
votes
1answer
126 views

$D_8$ as a derived subgroup

Every undergraduate student knows that there are (exactly) two non-abelian groups of order 8: Dihedral ($D_8$) and Quaternion ($Q_8$). The group $Q_8$ has many interesting properties; simple of them ...
2
votes
0answers
153 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
0
votes
1answer
64 views

Abstract orbits stabilizers

Consider $D_8$ acting on itself by conjugation. Find orbits and stabilizers for all elements of $D_8$. $$D_8=\{1,r,r^2,r^3,b,br,br^2,br^3\}$$ So far I have the orbits: {$1$}, {$r^2$}, {$r,r^3$}, ...
0
votes
2answers
197 views

How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^\star \to (\mathbb{Z}/154\mathbb{Z})^\star $ where $f(x)=x^5$?

How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^* \to (\mathbb{Z}/154\mathbb{Z})^* $ where $f(x)=x^5$? The group operation in this case is multiplication with ...
3
votes
1answer
137 views

Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$

I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$. I'm reading a paper with a result that implies ...
3
votes
1answer
117 views

Dihedral groups and normal subgroups

Consider $D_8$ ={$1,r, r^2, r^3,b,br,br^2,br^3$} and the subgroup, $H$={$1,r^2$} and $K$={$b$} of $D_8$ I really need some help with these particular problems. Show that $H\lhd$$D_8$, but ...
1
vote
4answers
76 views

The isomorphism from $S_3/\langle (123)\rangle$ to $\mathbb{Z}_2$.

Suppose that $N=\{(123), (132), \operatorname{e}\}$ and $N$ is normal in $S_3$. Show that the quotient group $S_3/N$ is isomorphic to $\mathbb{Z}_2$. What mapping should I use?
1
vote
2answers
150 views

Permutation representation of group described by $a_i^2=\theta^2=1, a_ia_{i+1}=\theta a_{i+1}a_i=a_{i+2}$.

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
4
votes
3answers
382 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
0
votes
4answers
175 views

Can a group of order 3000 be a simple group?

Can a group of order 3000 be a simple group? How about the case of a group of order 1000?
1
vote
2answers
122 views

Let H and K be subgroups of the finite group G and supposes $|H|^{2}>|G|$ and $|K|^{2}>|G|$. Prove $H \cap K$ has at least two elements

So I supposed $|H \cap K|>1$ $\Rightarrow |HK||H \cap K|> |HK|$ Which eventually implied that $\Rightarrow |H \cap K|>|G|$ Thus since G is a group, and H and K are subgroups then the ...
2
votes
1answer
646 views

Let G be a finite group and let H and K be subgroups of G. Suppose [G:K] and [G:H] are relatively prime. Prove G=HK [duplicate]

So I am rather confused on where to start this proof so all I've got is $[G:H \cap K]=[G:H][H:H \cap K]$ $[G:H \cap k]=[G:K][K:H \cap K]$ Thus that implies $[G:H][H:H \cap K]=[G:K][K:H \cap K]$ ...
3
votes
1answer
55 views

order of $\langle (123) , (234) \rangle$

As homework the teacher asks us to determine how many elements are there in $\langle (123) , (234) \rangle \subset S_4$ . I've started doing all the multiplications between the elements, and I've ...
1
vote
1answer
346 views

Computing $\bmod$s with large exponents by paper and pencil using Fermat's Little Theorem.

I'm having a bit of trouble computing $\bmod{mod}$s of large numbers using Fermat's Little Theorem. For example, how would you compute $7^{435627650}\mod 13$? The solution given is $435627650\mod ...