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

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3
votes
1answer
27 views

Intuition behind quotient groups?

I am having a hard time seeing the intuition behind quotient groups or rings. Intuitively, for a group, say Z/nZ would the quotient groups be the different sub groups of order 0 to n-1? Or how would ...
6
votes
6answers
73 views

What does it mean for something to hold “up to isomorphism”?

For example, to say that there are 2 such groups up to isomorphism such that the order of G is equal to $p^2$?
2
votes
1answer
47 views

Why do we have a basis?

A corollary that is in my book that I think is relevant to my question is: If E is an extension field of F, $\alpha \in E$ is algebraic over F, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ ...
0
votes
0answers
7 views

nilpotent algebraic groups in terms of extensions

Let $N$ be a nilpotent algebraic group over a field $k$. If $k = \mathbb{C}$ and $N$ is connected, one can write $N = U \times T$, where $U$ is a unipotent algebraic group and $T$ is a torus. Can ...
3
votes
0answers
26 views

Inner automorphisms as the kernel of a homomorphism

By a straightforward computation, it is not hard to show that the set $\operatorname{Inn}(G)$ of the inner automorphisms of a group $G$ is a normal subgroup of $\operatorname{Aut}(G)$, see for example ...
0
votes
2answers
30 views

A prime order group must be cyclic [duplicate]

I have a question about prime order group. This answer by amWhy says that: It follows that any group of order 5 (and any group of prime order) must be generated by a single element and is hence, ...
-1
votes
0answers
17 views

assume subgroup $H$ of $G$ such that $N$ is also a subgoup of $H$, then $ P_{G/N}(H/N) = P_{G}(H)/N$

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
1
vote
3answers
28 views

Why is it the smallest subfield containing F and $\alpha$?

Please take a look at the sentence in red: I understand that $\phi_\alpha[F[x]]$, is a subfield which contains $\alpha$, and F(we just need to evaluate $\phi_\alpha$ at the appropriate values). But ...
3
votes
1answer
56 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
0
votes
3answers
46 views

Does every finite field have a subfield $\mathbb{Z}_p$?

It seems that in the answers for my exercises in the book, the book uses that every finite field, has a subfield $\mathbb{Z}_p$. Is this true? They seem to use it in the answer for one exercise. But ...
-1
votes
1answer
28 views

Covering relation over functions

F is a group that includes all functions from N to N K is relation over F. For f,g ∈ F: (f,g) ∈ K iff ∀ n∈N, f(n)≤g(n). Obviously K is Partially ordered set and not Total Order. My problem is with ...
-2
votes
0answers
18 views

Let $N$ be a minimal normal subgroup of $G$, then $G/N$ is supersoluble? [on hold]

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
1
vote
1answer
19 views

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$?

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$ for each $r$, $1\leq r < p$? ($p$ is a prime) I guess I am just having a hard time understanding ...
1
vote
1answer
22 views

Is the unreduced Burau representation completely reducible?

To be specific, my question is about specializations $\beta \colon B_n \to GL_\text{n}\left( \mathbb C \right)$ of the unreduced Burau representation given by \begin{array}{cr} \beta \left( \sigma_{i} ...
1
vote
0answers
22 views

How can I find the Weights of a Subalgebra

I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group. For example, if we start with the 16-dimensional representation of ...
2
votes
1answer
36 views

Finite presentability of a group

Le $F$ be a free group of infinite rank (say countable rank) and consider a semidirect product $G = F \rtimes \mathbb Z$. Is it possible that $G$ is finitely presentable? If not, can we say something ...
9
votes
2answers
89 views

Prove that $x$ has order $5$.

let $ x \in G$ such that $(a^{-1})*(x^2)*(a) = x^3$ for some self inverse $a.$ Prove that $x$ has order $5.$ I don't know how to start this proof. Seems really difficult.
2
votes
2answers
57 views

Is every group of odd order isomorphic to a subgroup of $A_n$ for some $n$?

Is every group of odd order isomorphic to a subgroup of $A_n$ for some $n$? If not, what is a counterexample; if so, how can I prove it? Hints will be appreciated.
0
votes
0answers
41 views

Proving a group is $PSL(2,q)$ with $q>3$ odd.

How can I prove the following theorem If $G$ is a nonabelian simple group with Sylow $2$-subgroups being of order $4$ then $G=PSL(2,q) $ where $q>3$ is odd. with the help of this theorem: ...
2
votes
1answer
28 views

If $G$ is finite group that supersoluble then $G$ satisfy the maximal permutizer condition?

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
-1
votes
0answers
24 views

What is the permutizer of the Sylow 3 subgroup in $S_4$ ? [on hold]

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
3
votes
1answer
43 views

Classify $ \mathbb{Z}_9\times\mathbb{Z}_8\times\mathbb{Z}_8$/<(3,2,4)> according to the fundamental theorem of finitely generated abelian groups.

Since the order of $(3,2,4)$ is $12,$ the quotient group is of order $48.$ Now I have a problem. Consider $(1,0,1)+<(3,2,4)>.$ The element is of order $ 72,$ I think. But, it's impossible you ...
0
votes
1answer
44 views

If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$

These questions are really bothering me. Any help would be much appreciated. Let $p$ be prime a) If $(G, \oplus)$ has order $2p$, prove that every proper subgroup of $(G, \oplus)$ is cyclic b) If ...
0
votes
2answers
29 views

Prove that $(\mathbb{R^+} \times \mathbb{R^+}, \oplus)$ is a commutative group, where $(a,b) \oplus (c,d) = (ac, bd)$

Prove that $(\mathbb{R^+} \times \mathbb{R^+}, \oplus)$ is a commutative group, where $(a,b) \oplus (c,d) = (ac, bd)$. I had this written up but not sure if there needs to be more done. Thanks. Let ...
2
votes
0answers
24 views

Generate specific reduced words that “violate freeness”

Let $G$ be a group, and let $g_1,g_2\in G$ be nontrivial elements that do not commute. If $g$ and $h$ are not free as group elements, then the only a priori information this provides us is that there ...
1
vote
0answers
18 views

can say every group that satisfy in maximal permutizer condition then satisfy then permutizer condition

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
3
votes
1answer
37 views

Trying to show $|ab|$ divides lcm$(|a|,|b|)$

I'm trying to solve this Putnam problem. The problem is "show that for a finite group with $n$ elements of order $p$, where $p$ is prime, either $n=0$ or $p\: \vert\: n+1$." I'm trying to do this by ...
1
vote
1answer
48 views

Prove that $f$ is an onto function and a homomorphism function from $(\mathbb{Z} \times \mathbb{Z}, \oplus)$ to $(\mathbb{Z}, +)$

I have a lot of issues trying to figure out this problem. Any advice? Consider the two groups $(\mathbb{Z} \times \mathbb{Z}, \oplus)$ and $(\mathbb{Z}, +)$, where $(a,b) \oplus (c,d) = (a + c, b + ...
0
votes
1answer
22 views

What is the group $C_2^4$?

I'm trying to do a problem which asks me to show that a certain group is isomorphic to $C_2^4$. What is this group?
2
votes
1answer
31 views

Subgroups of every order dividing the order of the group imply the group is abelian?

Let $G$ be a finite group, denote $|G|=n$. I know about Cauchy theorem which states that if for a prime $p$: $p|n$ then there is $H\leq G$ with $|H|=p$. I also know that an abelian group $G$ have a ...
19
votes
1answer
247 views

Groups with “few” subgroups

If $G$ is a finite group of order $n,$ and the number of divisors of $n$ is $k,$ can $G$ have fewer than $k$ subgroups? A cyclic group $G$ of order $n$ has exactly one subgroup for each divisor of ...
1
vote
1answer
19 views

Group theory disjoint cycles

Let $a=(1 3 5)(1 5 6)(1 3 5)$ I had to write this as a product of disjoint cycles and got $(1 5)(3 6)$ which I believe is correct. Then figure out $a^{24}$ and $a^{25}$. Now $a^{24}$ is the ...
0
votes
1answer
48 views

Algebraic groups?

I have been doing group theory lately but I can not seem to find what I am looking for online (partly because I am not entirely sure what I am looking for). An example of one of the questions: If ...
0
votes
0answers
16 views

How to represent $B_4$ The braid group with $4$ strings non-pictorially?

How Do I represent $B_4$ The braid group with $4$ strings non-pictorially? What is the group presentation for $B_4$? It would seem the presentation is: $$B_4 = \langle ...
3
votes
2answers
58 views

Construct Group of Order 21 Without Semi Product

We have two possibilities, I know that one of the possibilities is the cyclic group$\frac{\Bbb{Z}}{21\Bbb{Z}}$. The other possibility as shown below with Sylow's theorems is $\Bbb{Z}_7 \times ...
2
votes
2answers
29 views

Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
3
votes
0answers
46 views

Normal subgroup of General linear group

What is the list of all normal subgroups of general linear group $GL_n(q)$? (n*n invertible matrix on finite field with $q$ elements) It is well known $SL_n(q)$ and subgroups of $Z(GL_n(q))$ are ...
1
vote
2answers
42 views

A question about direct product of subgroups.

Let $H$ and $K$ be two subgroups of a group $G$. Suppose that $H$ is normal in $G$ and $G/H\simeq K$. My question is when $G\simeq H\times K$? My guess is if $K$ is normal in $G$ and $G=HK$ then ...
3
votes
1answer
35 views

Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$ [duplicate]

Let $R$ be a ring. An element $r \in R$ is called nilpotent if $r^n=0$ for some integer $n \ge 1$. Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$. Proof. Recall ...
1
vote
2answers
37 views

What is $Q(x)$?

I do not really understand what $\mathbb{Q}(\pi)$ is here: Ofcourse we see that $\mathbb{Q}(\pi)$ is a field. But I have to "guesses" of what they mean, is one of them correct? 1. ...
3
votes
0answers
31 views

Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible. [duplicate]

Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible. Ok, si if $R$ is a ring with unity, then we have $R$ with $1 \ne 0$ We have ...
1
vote
1answer
19 views

Possibly ambiguous Cayley table construction

If $\{a,b\}$ with operation $*$ is to be a group, with $a$ the identity element, then what must the Cayley table be? I thought the following: \begin{array}{c|cc} * & a & b\\ \hline a & ...
10
votes
2answers
320 views

Does A5 have a subgroup of order 6?

I am trying to figure out if A5 has a subgroup of order 6. Rather than a yes/no answer I would prefer if someone could show me how they find their way to the answer. Below is my false attempt at a ...
2
votes
2answers
47 views

When can the rank of a submodule be bigger than the rank of the module itself?

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring. I am looking ...
1
vote
1answer
42 views

Show that every finite simple group G has a faithful irreducible representation

A representation $ \rho $ : G $ \rightarrow $ GL(V) is faithful if ker($ \rho $)={$ e $}. A representation is irreducible if it contains no proper invariant subspaces G is a simple group its ...
0
votes
0answers
25 views

A group which satisfies these conditions

I am looking for a group which satisfies the following conditions- $1.$ $G \le U_1(\mathbb{Z}G)$ , where $U_1(\mathbb{Z}G)$ is the set of normalized units of $\mathbb{Z}G$ i.e. $U_1(\mathbb{Z}G)$ ...
1
vote
1answer
45 views

Example of Abelian Group of order 2014 [on hold]

What are some examples of Abelian Groups of order $2014$ ?
0
votes
0answers
34 views

Fundamental Domain of ${\mathbb Z}^2$ to ${\mathbb R}^2$

Find a fundamental domain for the action of $\mathbb Z^2$ on $\mathbb R^2$ by translation A fundamental domain is the nodes $(0,0),(1,0),(0,1)$ and the edges which connect them Is there a better way ...
1
vote
2answers
35 views

Do groups of prime order have one subgroup?

So let's say that I have a group of order $p$, where p is prime; does that group only have one subgroup? I've look at the wiki article and it says there's a trivial and actual solution, so can we ...
1
vote
0answers
26 views

Decomposition of the Regular Q8 Module

For a worksheet we were asked to find the decomposition of the regular $Q_8$ module into a direct sum of simple modules. This isn't me asking for help on homework though, the problem is that I already ...