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.

learn more… | top users | synonyms

1
vote
3answers
163 views

What is a good way to think of Factor Groups?

I'm having a hard time thinking about factor groups. I just don't understand what notation like $\mathbb{Z}_{60}/\langle 12 \rangle$ means. Furthermore, when asked about giving the order $26 + \langle ...
0
votes
1answer
93 views

Automorphism of $Q_8$ [duplicate]

Is there anyone could help me to prove that $Aut(Q_8)=S_4$? Someone told me that there's an isomorphism between the rigid motions of cube and $Aut(Q_8)$, any ideas? Thank you!
2
votes
1answer
1k views

Order of subgroups and number of elements of order $3$ in a group of order $9$

Let $G$ be a group of order $9$. 1) State the possible orders of subgroups and elements in $G$. 2) Find the number of elements of $G$ of order $3$ in the cases where (a) $G$ is ...
0
votes
1answer
119 views

Every nontrivial subgroup $H$ of $S_9$ containing some odd permutation contains a transposition. [duplicate]

This is a true or false question. Apparently, it is false, but I don't follow. Clearly, if it contains an odd permutation, and an even/odd permutation is defined by the number of transpositions it ...
2
votes
1answer
40 views

Does the induced representation always contain a non-trivial representation

Let $H$ be a proper subgroup of a finite group $G$ - not normal. Does $Ind_H^G 1$ contain a non-trivial representation? The Frobenius character formula was my original approach, but I can't rule out ...
2
votes
1answer
114 views

How to calculate an index $(G:U)$

I have the group $G = \def\Z{\mathbb Z}\Z/9\Z$ and the subgroup $U = \{\bar 0,\bar 3,\bar 6\} \subseteq G$. My first question is, what mean the lines over the elements of the subgroup. And how can ...
2
votes
1answer
161 views

Calculate the Factor Group: $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$

I am attempting to understand and compute: $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$ I know $(0,2)$ generates $H = \{(0,0),(0,2),(0,4)\}$, which has an order of 3 because there are 3 ...
5
votes
1answer
489 views

Without Lagrange's theorem: If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$.

Prove the following without Lagrange's theorem: If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$. So this proof is pretty trivial if you have Lagrange's theorem, but ...
-1
votes
1answer
678 views

A subgroup of $S_n$ contains only even permutations or half of them are even. [closed]

Show that for every subgroup $H$ of $S_n$ for $n \geq 2$, either all the permutations in $H$ are even or exactly half of them are even.
1
vote
0answers
61 views

correspondence theorem question

If $G$ has order $12$ and $G'$ has order $6$, produced by elements $x$ and $y$ respectively, and $\phi$ maps $G$ to $G'$ which is defined by $\phi(x^n)=y^n$, how is the correspondence exhibited in the ...
1
vote
1answer
52 views

What are the cosets of this presentation?

I'm reading a book on algebra, and they give a presentation for $S_3$, with 6 elements $\{1, x, x^2, y, x y, x^2y\}$ as $$x^3 = 1,\quad y^2 = 1,\quad y x=x^2y$$ Now later in the book, there is a ...
2
votes
1answer
278 views

Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups

Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$. A. Find integer $n$ such that $U(n)$ contains ...
7
votes
1answer
152 views

character degree and solvability

There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here: Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in \operatorname{Irr}{(G)}$? Can ...
0
votes
1answer
95 views

What is the necessary and sufficient condition for this Cartesian product to be a cyclic group?

Let $m_1$, $m_2$, $\ldots$, $m_n$ be positive integers, and let $Z_{m_i}$ denote the group $\{0, 1, 2, \ldots, m_{i}-1\}$ under addition modulo $m_i$, for each $i = 1,2, \ldots, n$. Then what is the ...
1
vote
1answer
143 views

Show that $HK=\mathbb{Z}_n^\times$

Let $p$ and $q$ be distinct prime numbers and $n=pq$. Show that $HK=\mathbb{Z}_n^\times$ for the subgroups $H=\{[x]\in\mathbb{Z}_n^\times\mid x\equiv 1\pmod{p}\}$ and $K=\{[y]\in\mathbb Z_n^\times ...
10
votes
2answers
287 views

Can we conclude that this group is cyclic? [duplicate]

Let $G$ be a finite group. If, for each positive integer $m$, the number of solutions of the equation $x^m = e$ in $G$, where $e$ is the identity element, is at most $m$, then can we conclude that $G$ ...
2
votes
2answers
58 views

Subset of finite group

Let $G$ be a finite group and $A \subseteq G$. Suppose $2 |A| > |G|$. Prove that $G$ is equal to $AA$. $\left(AA = \{xy\mid x \in A\land y \in A\}\right)$
2
votes
1answer
151 views

Prove that product of a nilpotent group and a supersolvable group is a supersolvable group

Suppose that $G$ is a finite group, and $G = HK$, where $H,K \triangleleft G$, $H$ is a nilpotent group, $K$ is supersolvable group. Prove that $G$ is supersolvable group. Thanks. (The group ...
4
votes
2answers
121 views

Cohomology of finite groups with finite coefficients

I'm wondering if the group cohomology of a finite group $G$ can be made nontrivial with a nice choice of a finite $G$-module M. In other words, given a finite group $G$ and a number $n$, does there ...
6
votes
3answers
275 views

The group of invertible elements of $\mathbb F_{p}[x]/(x^m)$ is not a cyclic group.

I am stuck in a question about finite fields and would like to ask you for some help. Given an integer $m\geq 2$ and $p$ a prime number, show that $(\mathbb F_{p}[x]/(x^m))^{\times}$ (the group ...
-1
votes
1answer
87 views

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition? Here $S_9$ denotes the group of all permutations (i.e. bijections with itself) of the ...
0
votes
1answer
282 views

Normal Subgroups of a Nilpotent Group

Let G be a finite nilpotent group with order n. Is it necessarily true that for all divisors m of n, G contains a normal subgroup H such that ord(H)=m? Why or why not? I was able to show that G always ...
4
votes
1answer
90 views

Necessary and Sufficient conditions to generate $S_n$

I have a homework question that asks "Find necessary and sufficient conditions on $1 \leq i < j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$." Here is what I have done ...
12
votes
0answers
343 views

Values attained by $|G/Z(G)|$?

So I was working through some problems in a book on $p$-groups and noticed that $p$-groups have some really nice properties. So I started computing what the values of $|G/Z(G)|$ for $p$-groups. I ...
4
votes
1answer
94 views

On group theory terminology

Let $G$ be a finite group. Consider the next number $$m(G):=\min\{m\in\mathbb{N}\mid G\ \text{can be embedded into}\ S_{m}\}.$$ It is obvious that Cayley's theorem yields $m(G)\leq |G|$. My ...
1
vote
0answers
60 views

Upper central series in Coclass Theory.

It is proved by Aner Shalev, that for any finite $p$-group of coclass $r$(and sufficiently large order), there is some severe restrictions on lower central series $(\gamma_i(G))$. For instance, ...
1
vote
0answers
63 views

On some endomorphisms of finite groups of odd order.

Let $G$ be a group of odd order. It is known that if every central automorphism of $G$ acts trivially on the center, then $G$ is purely non-abelian, this amounts to saying that every central ...
1
vote
2answers
46 views

What does $2^H$ mean, where H is finite group?

From Henry Cohn paper: Definition 6.5. Let $H$ be a finite abelian group. An $H$- chart $\mathcal{C} = (Γ, A, B, C)$ consists of a finite set of symbols $Γ$, together with three mappings $A, B, C: ...
1
vote
0answers
101 views

Structure of finite abelian group

I am trying to solve this problem, and I think I should use the structure theorem for finite abelian groups, but i can't really figure this out. Let $G$ be a finite abelian group. Prove there is a ...
2
votes
1answer
2k views

If $G$ has only one subgroup of order $n$, then that Subgroup is Normal [duplicate]

How can I show that if some group $G$ has only one subgroup $K$ of order $n$, then $K$ is a normal subgroup? Would that mean that it only has one subgroup total? If so then I guess that makes sense.
4
votes
2answers
222 views

How can we determine associativity of a binary structure from its Cayley table?

Suppose $S$ is a finite set with a binary operation $*$ given by a Cayley table. While the commutativity of $*$ can be determined on the basis of the symmetry of the table across the upper-left to ...
3
votes
1answer
69 views

Alternate Proofs to this Question?

The question asks: "Does there exists a non-abelian group of order 2012?" My answer is yes, an example of which is the dihedral group $D_{1006}$. I'm curious, though, if anyone can give me a ...
-1
votes
1answer
93 views

What is the relation between these two subgroups of a finite cyclic group?

Let $G$ be a finite cyclic group of order $n$ generated by $a$. If $k$, $m$ are integers such that gcd($n, m$) = gcd($n,k$), then what is the relationship between the subgroups generated by $a^k$ and ...
6
votes
0answers
63 views

Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
1
vote
1answer
166 views

If $g$ has finite order $n$ show that $n$ is the least number such that $g^n$ is the neutral element

The order of the element $g$ is the size (cardinality) of the group $\langle g \rangle$. If $g$ has finite order $n$ show that $n$ is the least number such that $g^n$ is the neutral element. I can ...
2
votes
2answers
166 views

An abelian subgroup of a group of order 32:

I know that a group of order 32 with $|Z(G)|=4$ has an abelian subgroup of order 16 using GAP (Groups, Algorithms, Programming). Is there a way without using programming to show this. Any help will be ...
14
votes
1answer
344 views

How many subgroups does $\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 $ have?

How many subgroups does $H = \mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 $ have? $\mathbb{Z}_6$ has 4 subgroups (including itself), so the answer is at least $4^4$. But, ...
1
vote
2answers
75 views

all subgroups of $\mathbb Z_2^3$ including $(0,1,1)$

I want to write all subgroups of $\mathbb Z_2^3$ including $(0,1,1)$ I guess the only property that must hold is the closure. So every closed subset including $(0,1,1)$ are the solutions. Am I ...
3
votes
0answers
84 views

why does table look like this? - multiplicative table for Group

I have a Group $\mathbb{Z_2}[x]/q$ where $q(x)=x^3+x+1$ In my textbook, there a multiplication table: but I dont understand how the last $x+1$ comes. it should be $(x^2+x+1)*(x^2+x+1) \quad mod ...
0
votes
2answers
111 views

For group $\mathbb{Z_{18}^*}$, how do I find all subgroups

In my textbook, there is a cyclic Group $G=\mathbb{Z_{18}^*}$ which has the elements $$\{1,5,7,11,13,17\}$$ And its subgroups are $U_1 = \{1\}$, $U_2 = \{1,17\}$ and $U_3 = \{1,7,13\}$ How did they ...
1
vote
2answers
80 views

How do I know there are only 5 different groups of order 8? [duplicate]

How many different groups are there in order 8? And how do I know which groups they are? I mean, is there anyone can teach me to calculate them? I want a proof, thank you! They are $C_8$, $D_4$, ...
2
votes
6answers
2k views

Are two finite groups of the same order always isomorphic?

Are two finite groups of the same order always isomorphic? Some simple example would be great!
1
vote
0answers
128 views

Two questions in Isaacs' book Finite Group Theory

I am reading Isaacs' book finte group theory, and I have two questions. in page 90, there is a Wielandt's theorem (if $G$ has a nilpotent Hall $\pi$-subgroup, then all Hall $\pi$-subgroups of $G$ ...
-1
votes
1answer
119 views

Let $A$ be an abelian group. $A(p)$ is a p-group if $A(p)$ is finite.

Let $p$ be a prime number, let $A$ be an abelian group, define $A(p)$ to be the subgroup of all elements that have power of $p$ order, i.e. $x \in A(p) \iff x\in A \ \wedge \ p^ex = 0$, in additive ...
0
votes
1answer
112 views

Finitely generated abelian group isomorphic to infinite abelian group?

Lang's Algebra says that if an abelian group $A$ is free and finitely generated by $(x_i), i=1,\dots, n$ , then it is isomorphic to $\mathbb{Z}x_1 \bigoplus \cdots \bigoplus \mathbb{Z}x_n$, which is ...
0
votes
1answer
56 views

Two elements are in the same coset of $S$ iff their difference is in $S$

Assume $S$ is a subgroup of group $G$ How to prove this: Two elements are in the same coset of $S$ iff their difference is in $S$
0
votes
1answer
42 views

how do I find all Elements of a Group?

I am given a Group $\mathbb{Z_{11}^*}$. a multiplicative group. How do i find all elements of this Group?
1
vote
3answers
241 views

For any finite group, there is a homomorphism whose image is simple

This is for homework. The question asks "Show that, for any finite group $G$, there is a homomorphism $f$ such that $f(G)$ is simple." My thought was this. Since $G$ is finite, there are only a ...
1
vote
1answer
192 views

Prove that a group is a quaternion group.

The representation of the Quaternion group is $$\mathrm{Q} = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle.$$ Does this imply that as long as I have found a group with $4$ ...
0
votes
3answers
83 views

Exercise: product of transposition

How would I go about computing $$(1 2 3)\cdot(12)(34)$$ I know the definitions but I do not know how to apply them here. This is rather strange and odd-looking to me. I know I have to construct a ...