The study of symmetry: groups, subgroups, homomorphisms, group actions.

learn more… | top users | synonyms (2)

1
vote
1answer
75 views

The Conjugacy Classes Of Involution

what is the number of conjugacy classes of involutions in the two non-isomorphism double cover of Sn (2.Sn+) and (2.Sn-)
2
votes
1answer
28 views

Order of Factor Group of Direct Product

I'm trying to show that $G = (Z \oplus Z)/\langle(2,2)\rangle$ has infinite order (size). Can I just say that $\forall (a,b)\langle(2,2)\rangle \in G$, we can consider ...
0
votes
2answers
40 views

Group Homomorphisms (some questions about what I can and can't do)

Let $G$ be a group and fix some $g\in G$. Consider the map $\phi:\mathbb{Z}\rightarrow G$ defined by $\phi(n)=g^n$. Show that $\phi$ is a group homomorphism. Let $a,b\in \mathbb{Z}$. Then, ...
0
votes
2answers
114 views

Show $\phi$ is a homomorphism.

Let $\phi :\mathbb{R}\rightarrow \mathbb{S}$ be defined by $\hspace{150pt} \phi(r)=e^{2\pi ri}=\cos(2\pi ri)+i\sin(2\pi ri)$ Show that this is a group homomorphism. $\mathbb{S}$ is the unit circle. ...
-2
votes
1answer
37 views

Prove that the following are equivalen for abelian group

Let $(G, *)$ be a group. Prove that the following are equivalent: a. $G$ is abelian. b. $aba'b' = e$ for all $a,b \in G$. c. $(ab)^{2}$ = $a^2b^2$ for all $a, b \in G$.
1
vote
1answer
57 views

Let $G$ be the group $Z_2$. How would I define the following group, $K$?

$K$ consists of ordered pairs of elements of $G$, and the operation on K is $component-wise$ addition. For example: $(a,b) + (c,d) = (a+c, b+d)$. (These sums are taking place mod 2 of course) If I ...
1
vote
0answers
23 views

Discrete Logs and Generators Property

If given some $X$ that is $g^x$ and I want to find $x$ but cannot use $X$ directly does it follow that : $X*g \equiv g^\left(x+1\right)$ $ DLog(X*g) = x+1$ Therefore $x = DLog(X*g) - 1$? I tried ...
5
votes
2answers
75 views

Every finite Galois extension of $\mathbb{Q}_p$ has a solvable Galois group.

I have found the statement as in the title of this post on wikipedia, however, there is no reference for its proof. How does one prove it?
3
votes
0answers
46 views

Notation of quotient groups

So I'm determining the quotient group of $(E,+)$ in $(Z,+)$ where E is even int. and Z is int. I know sort of what is happening, I split the group $Z$ into evens and odds (2 sets) and as we are under ...
3
votes
2answers
1k views

If G is a group of order $pq$, where $p$ and $q$ are primes. How do I prove that any nontrivial subgroup of $G$ must be cyclic? [duplicate]

This makes sense since $p$ and $q$ are primes but I'm not sure how I can prove that the subgroup will be cyclic? Will the generator of the subgroup be $pq$?
1
vote
2answers
41 views

A group normal in $G$

I'm doing b) and c). If I assume $H$ is normal then $aH = Ha$ for all $a \in G \text{ and } N(H)$. If $N(H) = G$ then $G$ is somehow normal to itself...? Hints appreciated on both of these.
2
votes
1answer
71 views

Standard represention of $S_3$

I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in the ...
7
votes
3answers
185 views

What is the probability of product of two elements is desired element?

Let $G$ be a group with $n$ element. Fix $x\in G$. If you choose randomly two elements from $G$, what is the probability of $x$ being product of these two elements? At first, I thought answer was ...
2
votes
2answers
51 views

Determining whether $H$ is a normal subgroup of $S_4$

Do I really have to go through checking each and every left and right coset for this problem? Surely there must be a more efficient way that I don't know about?
0
votes
3answers
38 views

Questions on normalizer

I'm trying to do these and I'm a bit stuck on (a). I know that $a \in G$ and $H$ is a subgroup of $G$ so $aH = Ha \in G$. Notice that from this we can deduce that $H$ is a normal subgroup of $G$. ...
3
votes
0answers
67 views

$D_6$, regular hexagon.

Find a subgroup of $D_6$ where $D_6$ is the regular hexagon, with 12 symmetries. $$ D_6 = \{ e, r, r^2, r^3, r^4, r^5, a, ar, ar^2, ar^3, ar^4, ar^5\} $$ where $r^6 = e$. And the $r^n$ represent ...
0
votes
1answer
95 views

Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
-1
votes
3answers
56 views

Proving homomorphism

Problem. Let $G$ be a group. Define a function $f : G \longrightarrow G$ by for all $a \in G$, $f(a) = a^{-1}$. Prove that $f$ is a homomorphism if and only if $G$ is commutative. My attempt. ...
10
votes
2answers
496 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
1
vote
0answers
72 views

Is there an easy proof for the classification of $6$-transitive finite groups?

For the background, see the post: Classification of triply transitive finite groups Thanks to the classification of finite simple groups (CFSG), we know that if $G$ is a finite $6$-transitive ...
6
votes
1answer
192 views

How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?

The problem is as follows: Let $G$ be a group and $H$ a subgroup of $G$ (i.e., $H \le G$); let $x$ be any element of $G$ (i.e., $x \in G$). To prove that $[G:xHx^{-1}] = [G:H]$. I am able to ...
0
votes
1answer
50 views

Free action of finite direct product

Let's consider free action of finite abelian group $G = G_1 \oplus G_2$ on a manifold $X$. Is it true that $X/G$ is diffeomorphic to $(X/G_1)/G_2$?
1
vote
1answer
55 views

subgroup proof.

Prove that if $G$ is an abelian group, then $H =\{ x \in G\mid x^{2} = e \}$ is a subgroup of $G$. I did show that $H$ is close, associative, have identity and inverse element. Then my prof said I ...
0
votes
3answers
50 views

Proving that $G / N \cong (\Bbb R^\times, \cdot)$.

Let $G=(\Bbb C \backslash \{0\}, \cdot)$ and $N = \{a+bi \in \Bbb C : a^2 + b^2 = 1\}$. Prove that $N \lhd G$ and that $G / N \cong (\Bbb R^\times, \cdot)$. I already proved $N \lhd G$, but I'm ...
1
vote
1answer
34 views

Commutator Subgroup with Abelian Factor Group

I'm trying to show that if $Y$ is the commutator subgroup of a group $G$ (assumed to be normal in $G$ with $G/Y$ Abelian), and $N$ is a a subgroup with those same properties, then $Y$ is a subgroup ...
1
vote
1answer
42 views

Is $\langle S \rangle \cup \langle T \rangle = \langle S \cup T \rangle$ and $\langle S \rangle \cap \langle T \rangle = \langle S \cap T \rangle$?

Let $G$ a group, $S,T \subseteq G$ and $\langle S \rangle, \langle T \rangle$ the subgroups generated by $S$ and $T$. Is it true or false that $$\langle S \rangle \cup \langle T \rangle = \langle S ...
2
votes
3answers
41 views

If H is a subgroup of index 2 in $G$. Is it true that the set of all elements of $G$ that are not in $H$ is a coset of $H$?

We discussed this in class and it was seemingly just mentioned in passing. Is there a proof available to this?
3
votes
2answers
46 views

Proving that $[NH:N] | |H|$ and $[NH:N] | [G:N]$.

Let $G$ be a finite group, $N \lhd G$ and $H<G$. Prove that $NH < G$ and $[NH:N]$ is a divisor of $|H|$ and of $[G:N]$. What can we say about $N$ and $H$ if $(|H|, [G:N]) = 1$?. I already ...
2
votes
1answer
84 views

Prove that S consists of the quadratic residues mod (p) and T consists of the quadratic non-residues mod (p)

All the followings are $\bmod$ $p$ Let p be an odd prime. Suppose that the set $X = \{ 1,2, . . . , p-1\}$ can be written as the union of two nonempty subsets $S$ and $T$, where $S \neq T$, such that ...
0
votes
1answer
41 views

Explanation for the orders of subgroups and number of groups with these orders.

This was a question on my exam that was just given back and I need help understanding why both (a) and (b) and (c) are the answers they are. In each group listed below, give the orders of the ...
0
votes
1answer
52 views

An example of a morphism which does not preserve normality.

While doing an exercise, I was prompted to tell what I could say about the image of a normal subgroup under a morphism. While trying to reach a conclusion, I was able to demonstrate that if it is ...
0
votes
2answers
51 views

HK are subgroups of G

I'm proving that if $H$ is a subgroup of a group $G$ and $K$ is a normal subgroup of $G$, then $HK$ is a subgroup of $G$. I've tried pondering the fact that $HK$ is a subgroup of $G$ iff $HK = ...
0
votes
1answer
36 views

What is an example of an infinite group (say $G$) and a subgroup $H$ of $G$ which has index $2$?

So I know an infinite group has an order which, in a sense cannot be found. But would would be an example of a subgroup which has an index 2?
0
votes
3answers
67 views

Why does $aH = Ha \neq ah = ha$

For normal subgroups...How come $aH = Ha$ does not imply that $ah = ha$ for all $h \in H$? I'm showing that every subgroup of a commutative group is normal and I thought I had it on that one but ...
0
votes
2answers
53 views

What is the order of $f(x) = \frac{2}{2-x}$ as a permutation?

$f(x) = \frac{2}{2-x}$ defines a permutation of the set $A={\bf R}\setminus \{0,1,2\}$. What is its order in the permutation group $S_A$? I don't know how to find orders of functions. Please help!
2
votes
1answer
41 views

Abstract Algebra: Cosets

Find all of the distinct left cosets of <4> in Z18 and all the cosets of <4> in the subgroup <2> of Z18. So The distinct left cosets of <4> in Z18 are 0 + <4> and 1 + <4>. Do I ...
2
votes
2answers
95 views

If $[G:H]=2$, then G is Abelian

So suppose $G$ is a group, and $H<G$. If $[G:H]$=2, the $G$ is Abelian. I know I have use Lagrange here (okay, maybe I don't know, but I'm pretty sure). I DO know that $|G|=|[G:H]||H|$ which, if ...
12
votes
2answers
163 views

Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
1
vote
1answer
56 views

Constructing Topological Groups [closed]

In general, is there a way to construct topological groups? That is, given two topological groups $X$ and $Y,$ can I construct a topological group $Z$ using $X$ and $Y$ in said construction? I have ...
4
votes
2answers
117 views

Order of automorphism group

I have this tiny question that I just can't figure out: Let $G$ be the dihedral group of order 8. Show that Aut($G$) is a $2$-group. I know that there is a general way to calculate the order of the ...
1
vote
1answer
64 views

Minimum number of elements in a subgroup of a group of $15$ elements

Suppose a group $\mathcal G$ contains $15$ elements. $\mathcal H$ is a subgroup of $\mathcal G$ such that the minimum number of elements in $\mathcal H$ must not be less than $4$. What is the minimum ...
2
votes
1answer
108 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
1
vote
0answers
40 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
0
votes
1answer
77 views

Odd Permutations

Prove that the product of two odd permutations is even. I'm having a difficult time doing this in the general case. I have that if s is even, then $$\alpha = ...
3
votes
0answers
113 views

Classification of triply transitive finite groups

A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $X\setminus\{x\}$. Is ...
1
vote
1answer
29 views

Simple Probability/Group Theory Question

For any group G of order m and and a random m-tuple of elements from G, show that x1*...*xm is uniformly distributed over the elements of G or provide a counter example.
2
votes
1answer
301 views

G is group of order pq, pq are primes

Problem. Let $G$ be a group of order $pq$ such that $p$ and $ q$ are prime integers. I am to show that every proper subgroup of $G$ is cyclic. My attempt. What I know: Any element $a$ divides $pq$ ...
4
votes
2answers
519 views

Subgroups of the Klein-4 Group

Can anyone explain to me the subgroups of the Klein-4 group? I'm trying to view it this way: I want some groups that are not empty and $ab^{-1} \in H$, where $H$ denotes the subgroups I am looking ...
1
vote
1answer
38 views

seek results on existence of free subgroup with special property

Are there any known results of the following type? For $G=SO(3)$, does it contain a free subgroup $H=F_2$, the free group generated by two letters, such that $\forall g\in G, g^2\in H$ implies ...