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

learn more… | top users | synonyms (2)

5
votes
1answer
41 views

Characters of a Group: two definitions

If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms: $$\chi:G\longrightarrow\mathbb ...
1
vote
1answer
35 views

A question on the intuition of decomposition of the element of symmetry group

Any element of symmetry group $S_{n}$ can be decomposed as products of transpositions. Any m-cycle can be decomposed as m-1 transposition products. How should I think of this decomposition? Is there ...
0
votes
0answers
47 views

Wick rotation from $SO(2)$ to $SO(1,1)$: Howto

Both $SO(2)$ and $SO(1,1)$ are subgroups of $SO(2,\mathbb{C})$, one can choose a 1-parameter family of subgroups of $SO(2,\mathbb{C})_{\sigma(\phi)}\subseteq SO(2,\mathbb{C})$ defined by: ...
2
votes
1answer
50 views

Number of conjugacy classes in finite groups

Let $G$ be a finite group. Let $C_1,C_2,\dots,C_k$ be its conjugacy classes. We denote by $C_{j\ '}=\{g^{-1}|\ g\in C_j\}$ the conjugacy class inverse to $C_j$. Set $$a_{rst} = ...
-1
votes
0answers
36 views

Representations of group algebra and its centre

Are the irreducible representations of the algebra $Z(\mathbb{C}G)$ for a finite group G all irreducible representations of the algebra $\mathbb{C}G$, i.e. are the representations of the group algebra ...
1
vote
1answer
49 views

Topological group with discrete topology

Let $G$ be a topological group. I came to know that if I can show the existence of a homeomorphism of $G$ which moves only finitely many points of $G$, then $G$ has only discrete topology. How can I ...
10
votes
1answer
94 views

Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
3
votes
1answer
25 views

Triangle inequality and homomorphisms

Here is my situation: I have two homomorphisms $f$ and $g$ from a group $A$ into the complex numbers $\mathbb{C}$. I know that they are 'close' on a subset $B \subseteq A$. More formally there is an ...
0
votes
1answer
62 views

Equations modulo $143$

Let $x=11$ and $y=13$, and $z=xy=143$. (i) Show that $1$, $x+1$, $−1$ and $−(x+1)$ are the $4$ solutions of $n^2 \equiv 1\pmod z$. (ii) Find the coset of $U_z(2)$ consisting of solutions to $n^2 ...
1
vote
1answer
48 views

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$ . . .

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$. Then $U_g$ is a subgroup of itself. For every unit $c$ of $U_g$, show the coset, $cU_g = U_g$. Show that the product of the elements of ...
2
votes
2answers
46 views

How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
6
votes
1answer
189 views

Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
1
vote
1answer
47 views

I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
0
votes
1answer
56 views

Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
1
vote
0answers
15 views

“finite part” of an abelian pro-p group

I'm trying to understand the proof of the following: Let $G$ be an abelian pro-$p$ group, and let $N\leq _O G$ be an open subgroup such that $N\cong\mathbb{Z}_p$. Then $G\cong\mathbb{Z}_p\times T$ ...
1
vote
1answer
47 views

Group theory problem

I am asked to prove "Show that if $${e}<H_1<H_2<...<H_{n-1}<G$$ Is a subnormal series for a group G, and if the order of $H_{i+1}/H_i=s_{i+1}$, then G is of order $s_1 ...
1
vote
1answer
54 views

Artin Algebra 2.8.3 “Does every group whose order is a power of a prime $p$ contains an element of order $p$?”

I'm not sure whether or not my answer and proof for this question are valid. Could you point out any flaw? Let $G$ be an arbitrary group, an arbitrary element of $G$ be $g$ and $|G|=p^n$. Since a ...
3
votes
2answers
129 views

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
0
votes
0answers
18 views

Quotient splits in direct product

I have a quotient, say $$Pert(\mathcal{A} \oplus \mathcal{B}) / \ker(\phi) \cong Im(\phi) = Pert(\mathcal{A}) \times Pert(\mathcal{B}),$$ and I know that $$Pert(\mathcal{A} \oplus \mathcal{B}) \cong ...
1
vote
0answers
38 views

Show that an algebra is a linear algebraic subgroup in $GL(A)$, $A$ being a finite dimensional algebra over $\mathbb C$

Let $A$ be finite dimensional algebra over $\mathbb C$ with unit 1. Let $G$ be the set of all $g \in A$ such that $g$ is invertible in $A$. For $z \in A$ let $L_a \in$ End$(A)$ be the operator of left ...
2
votes
1answer
32 views

For a group $G$ acting transitively, is $G_{\alpha}$ contained in stabilizer of some block $\Delta$ if $\alpha \in \Delta$?

This question refers to permutation groups, in particular the primitive ones, and block systems. Let $G$ be a finite group acting on a set $\Omega$, and consider some partition $\Delta_1 \cup \ldots ...
5
votes
3answers
95 views

Quaternion Group as Permutation Group

I was recently, for the sake of it, trying to represent Q8, the group of quaternions, as a permutation group. I couldn't figure out how to do it. So I googled to see if somebody else had put the ...
1
vote
0answers
49 views

Does this subgroup of $\mathrm{SL}(2,\mathbb{C})$ have a a name?

The set of matrices $g$ characterized by $g=\begin{pmatrix}a&ib\\ ic&d\end{pmatrix}$, where $a,b,c,d \in \mathbb{R}$ and $ad+bc=1$, can be easily shown to be a subgroup of ...
9
votes
1answer
125 views

Are elements commutative when subgroups are?

Suppose G is a group and for every $H_1,H_2\le G$, $H_1H_2=H_2H_1$. Is $G$ abelian?
1
vote
1answer
141 views

Felix Klein's view on algebraic geometry

I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of ...
4
votes
2answers
110 views

A normal subgroup problem

Let $G$ be a group in which, for some integer $n>1$, $(ab)^{n}=a^{n}b^{n}$ for all $a,b \in G$. Show that $G^{(n)}=\{x^{n} \mid x \in G\}$ is a normal subgroup of $G$. $G$ could be easily ...
0
votes
0answers
15 views

semidirect product of cyclic and p-groups

How does conjugacy classes subgroups of a semi direct product group cyclic group of order (q-1) where q is a power of p, look like?
2
votes
2answers
72 views

why is it that the conjugate of a+bi is a-bi?

if a+bi is an element of a group, then its conjugate is a-bi, how can we prove this by using the fact that the conjugate of an element g of a group is h if there is an x in the group such that ...
3
votes
3answers
150 views

Basic Group Theory question

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ...
5
votes
1answer
53 views

Conjugate subgroups of $GL_n(K)$

Let $ K \subset L$ two fields, $G$ and $G'$ subgroups of $\mathrm{GL}_n(K)$. Assume that $G$ and $G'$ are conjugate in $\mathrm{GL}_n(L)$. Are $G$ and $G'$ conjugate in $\mathrm{GL}_n(K)$? ...
5
votes
1answer
73 views

Properties about order of a general group

For a group, $G$, is it true that $o(Z(G))\cdot o([G,G]) \leq o(G)$ where $Z(G)$ denotes the centre of $G$ and $[G,G]$ denotes the commutator subgroup of $G$?
0
votes
0answers
19 views

algebric subgroup of GL(A)

Let $A$ be a finite-dimensional algebra over $\mathbb C$. This means that there is a multiplication map $\mu : A \times A \rightarrow A$ that is bilinear. And let Automorphism Group of $A$ be ...
0
votes
2answers
59 views

Prove that any group of order 15 is cyclic. [duplicate]

Prove that any group of order $15$ is cyclic. I know that if order of group is a prime then the group is cyclic, but how to approach such questions?
5
votes
3answers
76 views

Pairs of $2\times 2$ matrices generating free groups.

The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, ...
1
vote
0answers
59 views

Is it “group axiom” or “group definition”?

Some text books of group theory use "group definitions" when introducing group, and some other text books use "group axioms". But it is obvious that terms "definition" and "axiom" are different. Which ...
1
vote
1answer
42 views

Show for each $c$, $\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$ is an abelian group under multiplication of congruence classes

Show for each $c$, the set $$\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$$ is an abelian group under multiplication of congruence classes.
4
votes
2answers
46 views

Density in $\mathbb{R}_{ +}$ of a subgroup of $\mathbb{Q}_{> 0}$?

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(1)=0$, $\phi(a.b) = \phi(a)+\phi(b)$, $\phi(a^{-1}) = ...
2
votes
3answers
55 views

Right-angled Artin groups are residually finite

I know that residual finitness of RAAGs (Right-Angled Artin Groups) follows from linearity, but does there exist a more direct proof, maybe simpler? EDIT: I added a proof based on cube complexes ...
0
votes
1answer
47 views

generators of groups from exact sequence

Suppose I have a middle term exact sequence of finitely generated abelian groups $G \longrightarrow H \longrightarrow K$. How do I get the generators of $H$ if I know the same for other two groups?
7
votes
4answers
357 views

What is a short exact sequence telling me?

Let's take a short exact sequence of groups $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$ I understand what it says: the image of each homomorphism is the kernel of the next one, so the ...
4
votes
1answer
81 views

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
-1
votes
1answer
81 views

Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
3
votes
1answer
96 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
2
votes
1answer
30 views

The minimum size of generating set of the external direct product

I have seen the following theorem here: Suppose that $A$ and $B$ are finite groups whose orders are relatively prime to each other, and the minimum size of generating set (i.e., the smallest ...
0
votes
0answers
59 views

Direct product of $G'$ and $Z(G)$ with some conditions

Let $G=G'\times N$ be a non-solvable group, such that $G/N$ is a non-abelian simple group and $N=Z(G)<C_G(N)$ is an abelian normal minimal $p$-subgroup of $G$. What can we say about $G$? In a ...
2
votes
1answer
81 views

Subgroup contained in all other subgroups

This is Problem 2.13.10 from Herstein, Topics in Algebra: Let $G$ be a finite abelian group such that it contains a subgroup $H_0 \neq (e)$ which lies in every subgroup $H\neq (e)$. Prove that $G$ ...
1
vote
1answer
56 views

Elements of orders $2k$, for $k\geq 5$ in a semidirect product

Let $G$ be a non-solvable group, $N$ be an abelian 2-subgroup of $G$ such that $N=C_G(N)$ and $G/N\cong Sz(8)$. Does $G$ has elements of orders $2k$, for $k\geq 5$?
0
votes
0answers
22 views

Factorization of parabolic subgroups.

Let $P$ be a parabolic subgroup of an algebraic group $G$. How to prove that $P = L_P U_P$? Here $L_P$ is the Levi of $P$ and $U_P$ is the unipotent radical of $P$. Thank you very much. Edit: I think ...
3
votes
2answers
52 views

Groups reluctant to have infinite subgroup

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
3
votes
3answers
222 views

What is the meaning of “fix” in field theory?

What is the meaning of "fix" in field theory? Example: I found a definition of field automorphism, A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational ...