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)

0
votes
2answers
22 views

Is $S_5$ isomorphic with the direct product $A_5 \times Z_2$?

Is $S_5$ isomorphic with the direct product $A_5 \times Z_2$? How i can check it?
0
votes
1answer
30 views

Actions of a finite group.

I've been playing around with this proof for a while and I can't seem to figure out where to go from here: I have that $M$ is a manifold, $G$ is a finite group (say, of order $n$), and the action of ...
1
vote
2answers
29 views

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$. I am trying to solve this but i need any ideas or hints to start,any help would be interesting.
1
vote
0answers
13 views

Connectedness of automorphism group of a variety

Let $Y$ be a proper, smooth, integral variety over an algebraically closed field $k$ of characteristic zero. Consider the automorphism group $Aut_k(Y)$ (a group scheme). Are there any natural ...
0
votes
0answers
27 views

The standard representation of $SO(n)$

This fact always bothers me. The group $SO(n)$ is defined as a set of $n\times n $ matrices, so this is a representation of the group in the first place. Yet, it seems that this representation has not ...
-4
votes
1answer
29 views

If $H<G$ is a normal subgroup of G,Show that the center $Z(H)$ of H is also normal subgroup of $G$

If $H<G$ is a normal subgroup of G,Show that the center $Z(H)$ of H is also normal subgroup of $G$ Any ideas for showing this?
2
votes
1answer
17 views

$ G/N $ is is a subgroup of a direct product of copies of the cyclic group $ C_{p-1} $ if $ p>2 $

Let $ G $ is soluble group and $ A $ be a unique minimal normal subgroup of $ G $. Then $ A $ is a elementary abelian group of a prime power. Let every chief factor of $ G/A $ has order $ 4 $ or a ...
0
votes
2answers
18 views

How to show #Hom$(C_a, G)=\{x\in G: x^a=e\}$?

I am willing to establish that $$\#\text{Hom}(C_a, G)=\#\{x\in G: x^a=e\}$$ where $G$ is finite group of order $n$ and $C_a$ is cyclic group of order $a$. I started like this: By first isomorphism ...
1
vote
4answers
53 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...
0
votes
0answers
15 views

The Second Homomorphism Theorem for groups

$G$ is a group. $N,H\le G$ $$$$ N is a normal sub-group of G. Want to prove $NH\backslash N\:$ isomorphic to $H\backslash N\cap H$ without using any homomorphism, but with understanding the quotient ...
0
votes
0answers
11 views

when a dihedral groups is nilpotent?

well i found this question answered in Is the dihedral group $D_n$ nilpotent? solvable? but the answer involves the sylow thm which i haven't studied. i'm studying bhattacharya and this question ...
0
votes
1answer
21 views

If $G_1/N \unlhd G/N$ then $G_1 \unlhd G$?

I want to show that if $N$ is a simple normal subgroup of a group $G$ such that $G/N$ has a composition series, then also $G$ has a composition series. I think I can finish the proof if I can only ...
0
votes
0answers
9 views

Let $G=Z_6 \times Z_9$ with $H=<(3,3)>$ and $K=<(0,1)>$. Find the order of $(1,0)$ in $G/H$ and in $G/K$

Let $G=Z_6 \times Z_9$ with $H=<(3,3)>$ and $K=<(0,1)>$. Find the order of $(1,0)$ in $G/H$ and in $G/K$ and determine the groups:$H \cap K$,$G/H$,$G/K$,$HK/K$,$HK/H$ and $G/H \cap K/H/H ...
-1
votes
1answer
18 views

Order of an element in a finite group - what does it tell me? [on hold]

I get the basics on how to calculate the order of an element in a finite group, but not sure why I want to. When I find the order what does it allow me to do or know about the group? Real world ...
1
vote
2answers
28 views

A group $G$ of order $32$ act in a set $X$ order $15$.Show that there is at least one element in set $X$ that remains stable under the action of $G$

A group $G$ of order $32$ act in a set $X$ order $15$.Show that there is at least one element in set $X$ that remains stable under the action of $G$ Any ideas and hints to show this?
0
votes
1answer
28 views

find all subgroups of G where: $0 \ne r \in \Re$ $G = <r>$

I need to find all subgroups of G where: $G \lt \Re$ $0 \ne r \in \Re$ $G = <r>$ $\Re$ is the group of real numbers and G is a subgroup. Edit : the operation is + I tried thinking about ...
2
votes
1answer
50 views

Possible order of $ab$ when order of $a$ and $b$ are known.

Let $a,b\in G$ be elements of a finite group $G$. We know $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. In dependence of $m$ and $n$ what are the possible values of ...
4
votes
1answer
37 views

Given a group of order $p^nq^2$ for two odd primes, prove that the commutator is a p group.

Given a group of order $p^nq^2$ for two odd primes $p > q$, prove that the commutator is a p group. To solve this question I need to prove that the commutator can't be of the orders $p^iq$, ...
0
votes
2answers
23 views

commutator (derived) subgroup of S3

how can i calculate it easily? i showed that the commutator group of S3 is generated by (123) in S3 using the fact that S3 is isomorphic to D6 and relation in D6 but that was tedious...are there any ...
0
votes
0answers
31 views

subgroups of the free product of a finite cyclic group and an infinite cyclic group.

How can I find all the subgroups of the free product $G = \mathbb{Z}*\mathbb{Z_2}$? I tried to answer this by looking at the subgroups of $\mathbb{Z}$ and $\mathbb{Z_2}$ separately. The subgroups of ...
0
votes
1answer
22 views

What should I do to tackle the following matrices calculation?

Through chapter 3 of Group Theory by Morton Hamermesh in part 3-6 (Equivalent representations; characters.) I stopped in some point. It's told "If we change the basis in the n-dimensional space $L$, ...
2
votes
0answers
17 views

Find the bigger possible order of element in the group $Z_2 \times Z_{36} \times Z_{10}$.Give an element in the group that has the order we found

Find the bigger possible order of element in the group $Z_2 \times Z_{36} \times Z_{10}$.Give an element in the group that has the order we found. How i can find the bigger order? i saw an example ...
1
vote
2answers
16 views

Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic

Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic Any ideas or hints for showing this?
2
votes
0answers
24 views

calculation $p$-Fitting subgroup

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $ that $ \vert A \vert = p^{a} $, $ p $ is prime. Let $ N =Fit(G) $, then $ N = O_{p}(G) $. Suppose $ F/N = ...
1
vote
1answer
38 views

Does such homomorphism exist?

$G$ is a group: $|G|=20$. Is there such a group G, for which the homomorphism $\tau :G-->Z_{10}$ exist?$$$$ The same question for: $\tau :G-->Z_{15}$ $$$$ I think that I should use here the ...
3
votes
2answers
37 views

Capable groups of order $32$ with GAP

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. How can I find all capable groups $G$ of order 32 with $|Cent(G)|=10$, where $Cent(G)$ is the set of all ...
0
votes
1answer
16 views

Subgroups of the unit circle under complex multiplication

Show that there are different subgroups of the unit circle which are isomorphic to ZxZ. I can show there are many subgroups of the unit circle which are Isomorphic to Z but I am having no idea for ...
0
votes
1answer
20 views

How to describe the transformation that changes French flag to Russian flag?

http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=Russia%2C%20France%20flags I presume it can be described two group operators, but I'm not sure how to come up with the formal description. ...
0
votes
1answer
14 views

Normalizer of Unipotent subgroup in General Linear group

Let $\mathrm{GL}(n,\mathbb{F}_p)$ be the general linear group over field of order $p$, and $\mathrm{U}(n,\mathbb{F}_p)$ be the subgroup consisting of upper triangular matrices with each diagonal entry ...
2
votes
1answer
42 views

When the elements of maximum order are $n$-cycles in $S_n$?

If the elements of maximum order in $S_n$ are $n$-cycles, then we can guess with few computations that $n$ must be at most $4$. How can we prove this? I tried the case in $S_{2n+1}$, the symmetric ...
1
vote
1answer
32 views

The concept of parity for members in a group

I was wondering if the concept of an even number has a construction within group theory? Furthermore does it have any application or further abstraction? For example; as we know that all even number ...
1
vote
1answer
34 views

$x$ and $g$ are elements of the group $G$, show that the order of $x$ is equal to the order of $g^{-1} xg$.

If $x$ and $g$ are elements of the group $G$, prove that $|x| = | g^{-1} xg|$. Deduce that $|ab| = |ba|$ for all $a,b \in G$. attempt: Let $|x| = n$ be the order of $x$ and $| g^{-1} xg| = m$ be the ...
0
votes
0answers
23 views

How many and which homomorphisms are there from $S_3$ to $Z_8$? After this find all the possible automorphisms of $Z_9$ [duplicate]

How many and which homomorphisms there are from $S_3$ to $Z_8$?After this find all the possible automorphisms of $Z_9$ Any ideas or help for finding this homomorphisms and automorphisms?
0
votes
1answer
33 views

Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
3
votes
0answers
21 views

Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ ...
5
votes
0answers
38 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
4
votes
0answers
15 views

Infinitely iterated square roots in groups

Let $G$ be a group. What are possible conditions on $G$ to ensure that there is no sequence $\{g_i\}_{i\in\mathbb Z}\subset G\backslash\{1\}$ such that $g_{i+1}=g_i^2$ for all $i\in\mathbb Z$? Does ...
1
vote
1answer
18 views

Cyclic Groups - $a^k = e \text{ iff } n|k$

I saw this proof in the book on Abstract Algebra. Here is part of it: Let $G$ be a cyclic group of order $n$ and $a$ is the generator of $G$. Then $a^k = e \iff n|k$ Proof: Suppose $a^k=e$. By the ...
0
votes
0answers
28 views

Prove that the center of a group with $385$ elements has an element of order 7. [duplicate]

Prove that the center of a group with $385$ elements has an element of order 7. By Cauchy's theorem I know that if I prove that $Z(G)$'s order is divisable by 7, we're done. So now I need to rule out ...
0
votes
0answers
16 views

The conjugate closure of a subset is a kernel of a permutation representation associated to a group action

Let $G$ be a group and $H$ be a subgroup of $G$. Let $A$ be the set of all left cosets of $H$ in $G$. We know that $G$ can acts on $A$ by $$g\cdot xH=gxH.$$ For any $g\in G$, define that ...
2
votes
0answers
22 views

Application of Bruhat decomposition in $\mathrm{GL}(n,\mathbb{F}_p)$

Consider the General Linear group $\mathrm{GL}(n,\mathbb{F}_p)$ over the field of order $p$, $B$ denote the subgroup consisting of upper triangular matrices, and $W$ denotes the subgroup consisting ...
1
vote
1answer
39 views

How to prove that O(Ng) | O(g)

I have this exercise: Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all $Ng\in G/N$, $$o(Ng)\mid o(g).$$ For now, without using the canonic homomorphism $\tau ...
1
vote
2answers
40 views

Homomorphisms from $\mathbb{Z}_p$ to $\mathbb{Z}_3$

For which odd values of $p$ can we find a non-trivial homomorphism from $\mathbb{Z}_p$ to $\mathbb{Z}_3$ ? Is there any method to find those homeomorphisms explicitly? I have no any idea to handle ...
2
votes
2answers
164 views

Group Action as permutations

I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . then the action may be viewed as permutation by $x ...
0
votes
0answers
31 views

Condition under which $HK$ is a subgroup

Suppose $G$ is a finite group and $H$, $K$ are subgroups. $H < N_G(K)$ is a sufficient condition for $HK$ to be a subgroup, but is is possible that $HK$ is a subgroup although neither $H$ nor $K$ ...
0
votes
1answer
36 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
0
votes
1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
1
vote
0answers
18 views

Enumerating functions modulo action on both the domain and codomain.

Let $Hom(A,B)$ be the set of functions from a finite set $A$ to a finite set $B$ and let $G_A \leq S_A$, $G_B \leq S_B$ be a subgroups of the permutation groups of $A$ and $B$. For $f,g \in Hom(A,B)$ ...
1
vote
2answers
28 views

Third isomorphism theorem, about quotients

I'm trying to understand the third isomorphism theorem statement, specifically the one in Wikipedia: https://en.wikipedia.org/wiki/Isomorphism_theorem#Third_isomorphism_theorem I'm stuck at the point ...
3
votes
0answers
33 views

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...