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

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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
21 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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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
8
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2answers
344 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
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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 ...
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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
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4answers
118 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 ...
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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: ...
3
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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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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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1answer
94 views

Importance of groups $(\mathbb R,+)$ and $(\mathbb Z,+)$ for Fourier series

I have heard that the groups $(\mathbb R,+)$ and $(\mathbb Z,+)$ are the most important groups for Fourier series. Why is this the case? Supposedly, it has something to do with the fact that for any ...
2
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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
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, ...
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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
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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
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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 ...
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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 ...
0
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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 ...
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 ...
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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 ...
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., ...
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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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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 ...
2
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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 ...
2
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3answers
136 views

Show every left coset is equivalent to the right coset

Assume that H is a subgroup of index 2 of finite group G. I'm asked to show that every left coset is a right coset of H. I first see that the number of left cosets of H in G is 2. I guess I am ...
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3answers
66 views

Prove that an abelian group of order 2n contains precisely one element of order 2

Assume that n is odd. I am allowed to also assume, without proof, that every finite group of even order 2n contains element of order 2. My proof is as follows: The lagrange theorem states that if ...
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0answers
30 views

can we express $D_{2n}$ as semi direct product?

can we express $D_{2n}$ as semi direct product? this is my answer:yes,if $D_{2n} =<a,b|a^2=e ,b^2=e ,aba^{-1}b^{-1}>$, then $<b> \triangleleft D_{2n}$ , on the other hand we have ...
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1answer
146 views

On groups with none of their quotient groups divisible [closed]

Does there exist a group $G$ that satisfies the following conditions: Any proper subgroup of $G$ is contained in a maximal subgroup. There is some $N\unlhd G$ such that $\frac{G}{N}$ is divisible. ...
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1answer
37 views

Isomorphism for group with 2 generators

I am trying to prove that a map $\rho: G \rightarrow GL_2(\mathbb{Q})$ is isomorphism. Here $G=\langle x,y|y^{-1}xy=x^2\rangle$ and $x$ and $y$ are sent to specific matricies say $x$ to $A$ and $y$ to ...
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1answer
33 views

How do I prove that a presentation of the free product is this? How does this given proof make sense?

Rotman - Introduction to the theory of groups p.390 Let $\{A_i:i\in I\}$ be a family of groups and let a presentation of $A_i$ be $(X_i|\Delta_i)$, where the sets $\{X_i:i\in I\}$ are pairwise ...
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2answers
47 views

Suppose $G$ is a group with exactly $8$ elements of order $3$. How many subgroups of order $3$ does $G$ have?

Suppose $G$ is a group with exactly $8$ elements of order $3$. How many subgroups of order $3$ does $G$ have?
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2answers
23 views

Mathematical expression for map from $[0,1]$ to $S^2$

A topological space is called arcwise connected if, for any points $x,y\in X$, there exists a continuous map $f: [0,1]\rightarrow X$ such that $f(0)=x$ and $f(1)=y$. Although it is intuitively ...
5
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3answers
46 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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3answers
57 views

Prove or disprove that $H = \{n \in \mathbb{Z}\,|\, n\, \text{is divisible by $8$ and $10$}\}$ is a subgroup of $\mathbb{Z}$.

Is my reasoning correct: since the integers divisible by $8$ and $10$ are only $1$ and $2$, can I say that $H=\{n \in \mathbb{Z}\,|\, n = 1, 2\}$. Now if $H$ is going to be a subgroup then it must be ...
3
votes
2answers
254 views

Homomorphism problem gone wrong

Okay, so I'm working on a homework problem in abstract algebra, and I have found the solution already, what I want to know is why my initial line of reasoning didn't work - i..e, what have I done or ...
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1answer
33 views

Assume G is a finite group and prove that the number of elements x in G s.t. $x^3=1$ is odd

My way of thinking is that since G is a finite group $x^3=1$ means that x has the order 3 or 1 and since both are odd it verifies our statement. Is this correct?
4
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1answer
32 views

Prove that two groups of functions are isomorphic

The two functions $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-1}{x}$ generate, with the operation of function composition, a group $G$ of functions. Prove that this group is isomorphic to the group $S_3$. I ...
1
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1answer
30 views

Finding an Abelian subgroup S5 of order 6

Let me just first clarify what I am looking for here. I am looking for a permutation $\sigma$ with five elements and its product will generate a permutation with 6 elements? I'm not so sure how that ...
1
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2answers
21 views

Clarification on finding another subgroup given the order of two existing subgroups

If we assume that G is abelian and that it has a subgroup of order 7 and another of order 11. If we were asked to find another subgroup of this group would we take the least common multiple of the ...
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1answer
63 views

Is this map an isomorphism or not?

Denote $F =\{f:\Bbb R\to \Bbb R\mid f \text{ is infinitely differentiable}\}$. Define $\varphi:(F,+)\to(F,+)$ by $[φ(f)](x) = \int_0^x f(t) dt $. Is this map an isomorphism or not? ...