A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

learn more… | top users | synonyms (2)

6
votes
3answers
299 views

Product of all elements of a finite group with an unique element of order 2

Well be with you, gentlemen. I have the following problem from Aluffi's Algebra: given a finite group $G$ with an unique element $f$ of order $2$, show that \begin{equation} \prod_{g\in G}g=f ...
0
votes
1answer
38 views

A problem involving centralizers and order of elements.

Let $a$ be an element of a group $G$ such that $|a| = 5$. Show that $C_G(a)=C_G(a^3)$, where $C_G(a)$ is the centralizer of $a$ in $G$. Also, find an element $a$ of some group $G$ such that $|a|=6$ ...
1
vote
2answers
66 views

Infinite Suzuki Groups

Often I found myself on a symbol like $Sz(F)$ where $F$ is an infinite field. What is the definition of an infinite Suzuki group? Are they linear groups? Where I could find some informations about ...
3
votes
1answer
53 views

Find a non-abelian group $G$ and a positive integer m such that for all $g,h \in G$, $(gh)^m = g^mh^m$and $(gh)^{m+1} = g^{m+1}h^{m+1}.$

Find a non-abelian group $G$ and a positive integer m such that for all $g,h \in G$, $$(gh)^m = g^mh^m$$ and $$(gh)^{m+1} = g^{m+1}h^{m+1}.$$ I can't find an example.
3
votes
1answer
66 views

Is the following group either a quaternion group or $D_8$?

Let $|G|=2^n$ and $Z(G)=G'=\Phi(G)$ where $\Phi(G)$ is the Frattini subgroup and $|Z(G)|=2$. Is $G$ necassarily either a quaternion group or $D_8$?
1
vote
1answer
48 views

A more swift method for Conjugation Classes

I am asked to find the conjugation classes of a group order n. I am aware what a conjugation class is and how to find it. My question: is there a quicker/more simple way to find the conjugation ...
0
votes
1answer
111 views

Equality of cosets implies equality of the original sets

Let $H_1$, $H_2$ be two subgroups of $G$ containing $K$, where $K$ is a normal subgroup of $G$. Then if $H_1/K = H_2/K$, prove that $H_1=H_2$. Attempt: Let $h_1K = h_2K$, for some $h_1 \in H_1$, ...
3
votes
1answer
92 views

Rings and Semi-simple rings

I'm failing to see which of the following are semi-simple rings, any help would be appreciated. $\mathbb{C}[X]$, the group ring $\mathbb{Q[Z]}$ and $\begin{pmatrix} \mathbb{Z} & \mathbb{Q}\\ 0 ...
5
votes
2answers
112 views

To prove , if Aut$ (G)$ is trivial then $x^2=e , \forall x \in G$

If for a group $G$ the only automorphism is the identity automorphism , then how do we prove that $x^2=e ,\forall x \in G $ ? I have only been able to prove that $G$ is abelian ; Please Help .
2
votes
1answer
96 views

character group of finite abelian group and induced homorphism

This is ex 5.7 of chapter 10 of artin's algebra (2nd edition) Suppose $\varphi:G \rightarrow G'$ is a homomorphism of abelian groups. Define an induced homomorphism $\hat{\varphi}" \hat{G'} ...
0
votes
1answer
86 views

Additive subgroups of the finite field GF($2^m$)

Consider the set $G=\left\{ {0,1,...,{2^m} - 1} \right\}$. The elements of this set can be viewed as the elements of GF($q=2^m$) with appropriate addition/multiplication operations. For example, GF(4) ...
5
votes
1answer
159 views

every group of order $105$ has a cyclic normal subgroup of index $3$ ?

Does every group of order $105$ has a cyclic normal subgroup of index $3$ ? (Please don't use Sylow theorems )
2
votes
1answer
76 views

A question in Abstract Algebra about cosets

I tried to solve this problem but without success: Let $H$ be a subgroup of a group $G$. Build an injective function from $G/H$ (the set of left cosets of $H$) to $H\setminus G$ (the set of right ...
0
votes
2answers
79 views

Two homomorphisms must be equal if $ \phi (a) = \psi (a) $

Given $G_1$, a cyclic group, and $G_2$ a group with $\phi$ and $\psi$ homomorphisms from $G_1$ to $G_2$ how can I show that $\phi=\psi$ if and only if $\phi(a)=\psi(a)$
0
votes
0answers
72 views

group of order $36$ abelian

Let $G $ be a group of order $36$.How to conclude whether it is abelian or not .I tried using Sylow's theorems by calculating the number of subgroups of order $4$ and $9$ but I am getting so many ...
4
votes
2answers
124 views

Let H be a proper subgroup of G of order prime $p^k$ and $N(H) = \{a \in G|aHa^{-1} = H\}.$Show that $N(H) \neq H.$

Let G be a group of order $p^k$ where p is a prime and k is a positive integer. Let H be a proper subgroup of G and $$N(H) = \{a \in G|aHa^{-1} = H\}.$$ Show that $$N(H) \neq H.$$ I think I need to ...
1
vote
1answer
221 views

Internal and Direct Product question, need help with explanation

I am asked to express a group G={1,7,17,23,49,55,65,71} under multiplication modulo 96 as an external and internal direct product of cyclic groups. However I also have an example to help with it, but ...
3
votes
1answer
86 views

Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?

My question is the one in the title. Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?
0
votes
0answers
54 views

Isomorphism from $Aut(\mathbb{Z}_2 \times \mathbb{Z}_4)$ to $D_8$

I want to show that $Aut(\mathbb{Z}_2 \times \mathbb{Z}_4) \simeq D_8$ (isometry group of square) I' have got an idea to show isomorphisms : $f: Aut(\mathbb{Z}_2 \times \mathbb{Z}_4) \rightarrow ...
0
votes
2answers
67 views

Infinite order group that has no nontrivial subgroup?

Is there any infinite order group that has no nontrivial subgroup? I guess there isn't, but I don't know how to approach.
1
vote
1answer
58 views

Transitive Actions

Given that transitive actions are in a bijection with conjugacy classes of subgroups of G, describe isomorphism classes of transitive actions for the following groups: $C_4, Z/8, C_2 × C_2, S_3$ Can ...
1
vote
1answer
151 views

Proof of the existence of inverse elements for a group

The group to be determined is defined as follows: $\{x\in\Bbb{Z^4}:x_1x_4=1+x_2x_3\}$ with $(x,y)\mapsto(x_1y_1+x_2y_3,x_1y_2+x_2y_4,x_3y_1+x_4y_3,x_3y_2+x_4y_4)$ $*$ denotes the operation. We have ...
1
vote
2answers
105 views

2 problems with group actions on a finite set.

I've got two problems, which I can't solve on my own so I ask you to help me a little, with some tips :) Prove, that every action of group ( order 9 ) on a set of 8 elements has a fixed point. Show ...
1
vote
2answers
52 views

Examples of group homomorphisms with isomorphic but not equal images

This may be a poor question. I am having trouble thinking of a pair of group homomorphisms: $\varphi, \Psi: G \rightarrow H$ between groups where $\varphi(G) \neq \Psi(G)$ but $\varphi(G) \cong ...
1
vote
2answers
135 views

Homomorphism preserves normality

let $\phi:G\rightarrow G'$ be a homomorphism, and let N' be a normal subgroup of G'. I want to show that $\phi^{-1}[N']$ is also normal subgroup of G. My work : since homomorphism preserves ...
0
votes
2answers
47 views

Determine, whether the following sets together with the mappings are groups or not

I'm a newbie in groups and currently working on some exercises to get familiar with the material. I have to determine, whether the following sets together with the mappings are groups or not. I have ...
-1
votes
1answer
93 views

Alternate Form of Correspondence Principle

Let G be a group and K a normal subgroup in G. Let f : G → G/K be the canonical epimorphism given by x 􏰁→ xK and L a subgroup of G/K. Then (1) There exists a subgroup H of G. H contains K, and ...
0
votes
1answer
76 views

Is infinite product of Z a group?

Having the usual coordinate-wise addition, does infinite product of $\mathbb{Z}$ forms a group? $a,b\in \prod ^\infty \mathbb{Z}$ $a=(a_1,a_2,...)$ $b=(b_1,b_2,...)$ $a\circ ...
2
votes
2answers
214 views

Groups of order 24.

I supposed $n_3=4$ and $n_2=3$, and then I made $G$ act by conjugation on $Syl_3 (G)$. I want to show that $G\cong S_4$ (looking at all order 24 groups here ...
2
votes
1answer
81 views

Breaking RSA if small subset invertible

I am trying to solve a problem which states that one can invert RSA if a small subset of the cipher text are invertible, the problem is as follows: Given a function which can invert the RSA ...
0
votes
2answers
53 views

Does there exist a non-trivial homomorphism from $M_2(\mathbb Z/3\mathbb Z)$ to $Aut(\mathbb Z / 24\mathbb Z)$?

What I've first done is show that $Aut(\mathbb Z / 24\mathbb Z)$ is isomorphic to $\mathbb Z_2\oplus\mathbb Z_2\oplus\mathbb Z_2$ due to it having 8 members and the greatest rank out of all of them is ...
2
votes
1answer
88 views

Possible number of groups of order N

When is the number of groups of some order $n$ greater than $n$? For example, lets say this happens at $n=3$, then that would mean that there are more groups of order 3 than 3.
2
votes
1answer
41 views

A commutator relation

I hope the following is not trivial, Let $H$ be a subgroup of $G$ s.t. $[H,G]\leq Z(G)$ then can we say that $H$ is normal ? I think we can not but I could not find counter example. Any counter ...
1
vote
2answers
33 views

Existence of Unique Homomorphism Implies Generating Set

The following question is taken from "An Invitation to General Algebra and Universal Constructions", p. 23 Ex. 2.1.2 (available online here). Let $G$ be a group and let $\{a,b,c\}\subseteq G$ such ...
0
votes
2answers
318 views

The sum of the elements in a field of at least three elements is 0

This statement seems so simple yet I don't quite know how to start with this proof in a substantial way. Can anybody help me here?
1
vote
1answer
35 views

Why the homomorphism from g acting on a to left coset of stabilizer of a is surjective?

Suppose $b = g \cdot a$. Then $gG_a$ is the left coset of $G_a$. The map $b = g \cdot a \rightarrow gG_a$ is a map from $C_a$ to the set of left cosets of $G_a$ in $G$. Dummit says this map is ...
0
votes
1answer
58 views

Nilpotent torsion-free Groups with a fixed Soluble length

Let $s$ be a natural number. Is it possible to find, for each $n$ natural number greater than some arbitrary constant, a torsion-free group whose nilpotency class is $n$ and soluble length is $s$?
3
votes
1answer
133 views

Intuitive meaning of left and right cosets?

What is the difference between left and right cosets? I know their definition, but what I am seeking is the intuition behind left and right coset. I used to think of cosets as slicing a group (since ...
0
votes
0answers
33 views

If $H$ and $K$ are subgroups of a group G, and $a, b \in G$. How do I prove the following relations hold? [duplicate]

Either $Ha \cap Kb = \emptyset$, or $Ha \cap Kb = (H \cap K)c$, for some $c \in G$. I tried supposing $Ha \cap Kb \not= \emptyset$ but I can't see how it follows that $Ha \cap Kb = (H \cap K)c$.
1
vote
0answers
57 views

Showing well-definedness of a group operation

Let p be an odd prime integer, and let $r \in \mathbb{Z}$ with $1 \leq r < p$. Let G be a cyclic group of order p generated by g, and let K be a cyclic group of order p-1 generated by k. For ...
1
vote
1answer
25 views

Show that a certain normal subgroup or a product is abelian

Let $A$ be a normal subgroup of $G\times H$. If the identity of $G$ is $1_G$ and the identity of $H$ is $1_H$, $(x,y)\in A$ has the property that $x\ne1_G$ and $y\ne1_H$ unless $(x,y)=(1_G,1_H)=1_A$. ...
4
votes
4answers
380 views

Alternate proof of Schur orthogonality relations

I am trying to find an alternate proof for Schur orthogonality relations along the following lines. Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$. Let $V$ ...
1
vote
2answers
106 views

Order of group from its presentation

Say we want to determine the order of a group generated by $x$ and $y$ who satisfy $x^2y = xy^3 = 1$. Ok so it would be nice to know the order of $x$ and $y$ respectively. We can readily conclude ...
9
votes
0answers
64 views

A hard question on surjective group homomorphism [duplicate]

Say $G$ and $H$ are finite groups, and there exists a surjective group homomorphism from $G × G$ to $H × H$. Must there exist a surjective group homomorphism from $G$ to $H$? I have no idea how to do ...
2
votes
1answer
41 views

A specific group lacking Følner sequences

How does one go about proving that the free group $<a,b,a^{-1},b^{-1}>$ lacks any Følner sequence?
1
vote
1answer
70 views

Find every homomorphism from $C_{10}$ to $S_3$

Find every homomorphism from $C_{10}$ group to $S_3$. Find every homomorphism from $S_3$ to $C_{6}$ I know that : $S_3$ is permutation group of $3!=6$ elements, and $C_{10} = ...
4
votes
0answers
92 views

What branch of math is this?

In this paper: http://arxiv.org/pdf/hep-th/0505016v1.pdf what are the branch(es) of math being used? The unnumbered eq. on the top of page 3 and eq. (7) are good examples. All I've been able to figure ...
1
vote
1answer
92 views

If G is finite p-group and H is a subgroup, show that there is a composition series that contains H.

Let $p$ be a prime number, suppose $G$ is a finite p-group and let $H$ be a subgroup of $G$. Show that there is a composition series that contains $H$. I have already shown that if $G$ if a finite ...
0
votes
1answer
99 views

Subgroup with finite index contains a normal group [duplicate]

Let H be a subgroup of finite index of an infinite group G. Prove that G has a normal subgroup of finite index which is contained in H. I am not sure how to start on this problem, and would ...
0
votes
0answers
31 views

Normal series and refinements

I am trying to show that no nontrivial refinements exists in the following series but haven't been able to make any progress: $\{1\}\unlhd Z(SL_2(\mathbb{R}))\unlhd SL_2(\mathbb{R})\unlhd ...