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

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2
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1answer
35 views

Cyclic group generators

My question is: Can you find a cyclic group with n generators? I know that zero (or any other identity element for that matter) is included, so there would be for $Z_n$ at most n-1 generators. ...
1
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2answers
60 views

Abstract Algebra : Subgroups

I've been studying about subgroups and I encountered an example with answers and does not have explanation how it is derived and I need help to understand it. Here is the example: Example 1.4.20 ...
0
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1answer
85 views

Finding the center of a group with MAGMA

Given a finitely presented group $G = \langle S | R \rangle$, is there a command in MAGMA that computes the center of G ?
2
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1answer
30 views

Existence of an open normal subgroup of a neighborhood of 1 in a compact Hausdorff and totally disconnected topological group

Let $G$ a compact Hausdorff and totally disconnected topological group. Then every neighborhood of 1 contains an open normal subgroup of finite index in $G$. I need this lemma to prove that every ...
1
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1answer
35 views

Group Theory Formatting Question

I apologize if this was posted already somewhere. I looked but had trouble describing it. My only question is what the overline means in this situation. I do not need help with the actual problem. ...
1
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3answers
39 views

Alternating groups, specifically $A_6$

My question is what are the possible order of $A_6$? And how would I show I get $\frac{6!}{2}=360$. Any tips? I know that $A_6$ is the group of even permutations on six elements. I also know that ...
2
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2answers
36 views

Find all kinds of homomorphisms from $(\Bbb Z_n, +)$ to $(\Bbb C^*, \times)$.

Find all kinds of homomorphisms from $(\Bbb Z_n, +)$ to $(\Bbb C^*, \times)$ and from $(\Bbb Z, +)$ to $(\Bbb C^*, \times)$. Explain why they are the complete collection. My intuition is: 1) we ...
1
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1answer
30 views

Find **Pontrjagin dual.** of $\Bbb Z_n$ & $\Bbb Z$

Let $G$ be a group. Consider the set $Hom(G,\Bbb C^*)$ of homomorphisms from $G$ to $\Bbb C^*$. Define a binary operation $+:Hom(G,\Bbb C^*)\times Hom(G,\Bbb C^*)\to Hom(G,\Bbb C^*)$ s.t ...
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1answer
25 views

Stabilizer of a doubly transitive is maximal?

Is it true that if $G$ is a group acting $2$-transitively on a set $X$ , then if $x\in X$, then $G_x$ (stabilizer) is maximal in $G$. I think it must be true as a conclusion of $2$ theorems, as ...
2
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2answers
32 views

Subgroup generated by $G-H$ , $H$ is a subgroup of $G$

Prove that group generated by $G - H$ equals $G$, where $H$ is a proper subgroup of $G$.
1
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2answers
58 views

Show that there is no epimorphism from ($\mathbb{Q}$, +) to ($\mathbb{Z}$, +)

I can't figure out why such an epimorphism cannot exist. I think the fact that a bijection exists between the $\mathbb{Q}$ and $\mathbb{Z}$ throws me off a little. What is it about having a ...
0
votes
1answer
20 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
0
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0answers
26 views

Semidirect product of rings?

I'm trying to understand a paper. We are given a $\mathbb{C}[m^\pm, l^\pm]$-module $M$. Then we want to extend the module structure to $\mathbb{C}[m^\pm, l^\pm]\rtimes \mathbb{Z}_2$. The action of $s ...
0
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0answers
26 views

Are these natural actions of $SL(3,2)$ over $\{1,…,7\}$?

Let be $G=SL(3,2)$. I know that $\alpha=\left(\begin{array}{ccccccc} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{array}\right),$ $\beta=\left(\begin{array}{ccccccc} 1&0&0\\ ...
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0answers
39 views

inner automorphisms of non-abelian simple groups

Let $G$ is non-abelian and simple group. Let $I ={\rm Inn}(G) \cong G$, $A = {\rm Aut}(G)$ and $B = {\rm Aut}(A)$. Since $Z(A)=1$, we have $A \cong {\rm Inn}(A)$, so we can identify $A$ with the ...
3
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1answer
52 views

Meaning of natural action

What is the meaning of the assertion :'the group $G$ acts in a natural way on the set $S$'? I don't understand the meaning of 'natural'. thanks!
0
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1answer
76 views

$G$ infinite abelian group with $[G:H]$ finite for every non trivial subgroup $H$ , to prove $G$ is cyclic

Let $G$ be an infinite abelian group such that for any non-trivial subgroup $H$ of $G$ , $[G:H ]$ is finite ; then how to prove that $G$ is cyclic ? Please don't use any structure theorem of abelian ...
1
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1answer
49 views

Every subgroup is a union of cyclic subgroups of its group.

True or false? Every subgroup H of a group G is a union of cyclic subgroups of G. I think it is false,but cant think of counter example
1
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1answer
22 views

Prove that an element of a group is in a one-element conjugacy class if and only if it commutes.

I am trying to prove that an element $g$ of a group $G$ is in a one-element conjugacy class if and only if it commutes with all of $G$. The "only-if" direction is easy: Suppose $g$ commutes with all ...
0
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0answers
50 views

A question on p-groups, and order of its commutator subgroup [duplicate]

$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$. $\textbf{TRY- }$ If $P=Z(P)$ it is true. Now let $n > 1$, then If I see $P$ as a nilpotent ...
3
votes
2answers
78 views

Sylow's Theorem Explanation

Can someone explain it to me? I've been working out of Galian's Contemporary Abstract Algebra this semester, but came into possession a copy of Dummit and Foote's book, which I am aware is ...
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0answers
11 views

Can the generators of SU(4) be divided into 3 distinct subgroups?

Given the 15 generators (see last page) of SU(4), is it possible to divide them into 3 subgroups of 5 elements?
9
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4answers
116 views

If permutation matrices are conjugate in $\operatorname{GL}(n,\mathbb{F})$ are the corresponding permutations conjugate in the symmetric group?

There is a standard embedding of the symmetric group $S_n$ into $\operatorname{GL}(n,\mathbb{F})$ (for any field $\mathbb{F}$) that sends each permutation in $S_n$ to the corresponding permutation ...
2
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4answers
34 views

Generators of a product of finite abelian groups

Let $n_1,...,n_r$ be positive integers. Consider the group $$G={\bf Z}/n_1 {\bf Z} \times \cdots\times {\bf Z}/n_r {\bf Z}$$ When does a given element $(k_1,\cdots,k_r)$ generate $G$? Obviously ...
14
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3answers
552 views

Can a group have more subgroups than it has elements?

I'm looking for a group for which the number of subgroups is more than the number of elements in the group! I tried a few possibilities - it can't be cyclic, and I think we'll have to consider group ...
1
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1answer
25 views

Show $a\in G$ is contained in $Z(G)$ iff $Z(a)=G$ for center and centralizer? [closed]

The center of a group $G$ is defined as the set $Z(G):= \{a\in G\mid \forall b\in G : ab=ba\}$ and the centralizer of an element $a\in G$ is defined as the set $Z(a) := \{b\in G\mid ab=ba\}$. How can ...
8
votes
2answers
342 views

Group with exactly 2 elements of order 10.

Does this exist? I dont think it does. For any cyclic group the totient function of 10 is 4, so there is 4 of them. But also if one element is of order 10, say $a$, then $a^3$, $a^7$ is also of ...
3
votes
0answers
45 views

Automorphism on Symmetric Group and Transpositions

I've been looking into the group of automorphisms of the symmetric group $S_{n}$ for when $n > 6$. Something which is claimed frequently is that if an automorphism sends a transposition to a ...
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0answers
72 views

Is this problem still open or solved?

Problem- Let $n\geq$ and let $T$ be the set of all permutations in $S_n$ of the form $t_k=\prod_{1\leq i\leq k/2}(i,k-i)$ for $k=2,3,4.....(n+1)$. Then find the least integer $f_n$ such that ...
1
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1answer
38 views

Hall's Subgroup Theorem

I am currently looking into the extension of Sylow's theorems, namely through Hall-$\pi$-subgroups and Hall's Theorem. I currently have the theorem as; Let $G$ be a finite solvable group a $\pi$ be ...
2
votes
2answers
41 views

Centre of the group S4

Quite a simple looking question guys, Find the centre $Z(S_4)$ of $S_4$. The previous part asked me to find centralizers for $S_4$. I note that $Id$ is the only element contained in everything so I ...
2
votes
1answer
54 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
0
votes
1answer
37 views

Let $\alpha$ and $\beta$ be disjoint cycles. Prove for every positive integer n, $(\alpha\beta)^n=\alpha^n\beta^n$

Let $\alpha$ and $\beta$ be disjoint cycles. Say $\alpha = (a_1a_2...a_s)$, $\beta=(b_1b_2...b_r)$. Prove for every positive integer n, $(\alpha\beta)^n=\alpha^n\beta^n$ My proof is as follows: ...
2
votes
1answer
55 views

Is it Possible to Add or Multiply Groups?

I came across a GRE Mathematics Subject test question that said the following: "Find the characteristic of the ring $Z_2 + Z_3$." The explanation of the question starts with the statement that ...
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1answer
50 views

Abelian group with a finite number of subgroups is finite [closed]

Show that if $G$ is a abelian group with a finite number of subgroups then $G$ is finite.
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2answers
36 views

Sample groups Klein V, $Z_4$, $S_3$, $D_4$.

These four groups: Klein V, $Z_4$, $S_3$, $D_4$ were probably the most interesting examples used to solve for counterexamples so far. They're so useful that I can most likely guess that one of the ...
0
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1answer
33 views

Proof of isomorphism between $D_{2n}$ and $D_n \times Z_2$ for $n$ odd

If we define a function $\phi : D_{2n} \rightarrow D_n \times Z_2$ for odd $n$ and we want to show that it is an isomorphic function, I am not very sure how to do it. We know that $D_{2n} = \{e, r, ...
0
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0answers
15 views

Monotone actions of the infinite cyclic group

For reasons which are too long to explain, I grew interest in the theory of monotone actions of partially ordered groups, by which I mean monotone functions $G\times P\to P$ satisfying the axioms of ...
3
votes
2answers
39 views

Showing $\text{Aut}(\mathbb{Q})$ is the trivial group and calculating $\text{Aut}(\mathbb{Q}(\sqrt{2}))$

For a field $K$ let $\text{Aut}(K)$ denote the group of all automorphisms $f:K\to K$. How can one show that $\text{Aut}(\mathbb{Q})$ is the trivial group and how to calculate ...
2
votes
2answers
97 views

Let $\sigma \in S_n∖A_n$ ($\sigma$ not in $A_n$), prove that the order of $\sigma$ is even.

Given $\sigma \in S_n\setminus A_n$, prove that the order of $\sigma$ is even. I feel that I have a way to prove it: Since $\sigma \notin A_n$, the sign of $\sigma$ is $-1$. This implies that ...
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2answers
45 views

Proving that the set $([0,1[, \ast)$ is a group

Let in the set $G=[0,1[$ the operation $\ast$ defined by $a\ast b=a+b-[a+b]$ with $[a+b]$ the integer part of $a+b$, i.e, $a\ast b$ is the decimal part of $a+b$. I need proof that $(G, \ast)$ is a ...
2
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2answers
95 views

Can a group be a union of three subgroups?

I am being asked to show an example of when this fails or prove it rigorously. I am thinking of using an example to disprove the claim that a group can be a union of three subgroups. However I am ...
4
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1answer
40 views

Why Mathieu group M11 is sharply 4-transitive?

I am studying Steiner system and Construction of Mathieu groups from automorphism of some Steiner system.Mathieu group M11 is automorhism group of S(4,5,11) Steiner system. I am not able to understand ...
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0answers
31 views

The irreducible representation of $S_n$ of degree $n-1$ [duplicate]

So I understand it's not hard to show that the standard $(n-1)$-dimensional irreducible representation of $S_n$ is the only irreducible representation of $S_n$ of degree $n-1$ using characters/Young ...
0
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1answer
19 views

A transitive subgroup H of $S_n$

I have to prove the following: A transitive subgroup $H$ of $S_n$ is primitive if and only if for all $x\in X$, we have that $H_x:=\{ \sigma \in H : \sigma(x)=x \}$ is a maximal subgroup of H (i.e., ...
5
votes
3answers
45 views

Connectedness of $O(3)$ group manifold

A topological space is said to be connected if it cannot be written as $X=X_1\cup X_2$, where $X_1,X_2$ are both open and $X_1\cap X_2=\emptyset$. Otherwise, X is called disconnected. Is it wrong to ...
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2answers
55 views

Two abelian groups with the same order are isomorphic? [closed]

True of false: if G and H are two groups with the same order and both are abelian, then they are isomorphic.
2
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1answer
249 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
1
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1answer
46 views

Is a group structure is need for this analysis

We know that $ℤ/nℤ$ is a finite set, then it is possible to find a bijection $$θ:ℤ/nℤ→T={1,2,...,n}$$ where $T$ is a finite part of natural numbers ℕ. Let us consider the set $G=ℤ^{r}×ℤ/nℤ$, ...
2
votes
1answer
31 views

prove that $D_8 \cong C_2 \wr C_2$ .

prove that $D_8 \cong C_2 \wr C_2$ . $\wr$ is wreath product and it is using in place of $C_2 \ltimes (C_2 \times C_2)$. here is my answer : suppose $K=C_2 \times C_2$ and $C_2 \cong \langle \sigma ...