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

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A question on left cosets of distinct subgroups and index [duplicate]

Let $H_1 , H_2 , ... , H_k $ be subgroups of $G$ and $x_1,x_2,... ,x_k$ be elements of $G$ such that $G=\cup_{i=1}^k x_iH_i$ , then how do we prove that some subgroup $H_i$ has finite index in $G$ ?
3
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1answer
26 views

On non-trivial normal subgroup(s) of $A(S)$ , where $S$ is infinite

Let $A(\mathbb R)$ be the permutation group of $\mathbb R$ , is this group simple ? In general for an infinite set $S$ , how may we determine whether $A(S)$ has any non-trivial normal subgroup or not ...
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2answers
44 views

What do $\Bbb N^*$ and $\Bbb Z(p^n)$ mean in this paper?

There is a theorem: in this paper: http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ78_01%2FS1446788700015548a.pdf&code=2ffd5c5100675caf83c2e95bce65491e But there is no explanation ...
0
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1answer
51 views

Normal subgroup of a finite group

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?
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1answer
35 views

what are the p-orbits in the decomposition of Σ into p-orbits.

At the bottom in the proof of slow 2, what's the meaning of "restrict the action of G on Σ to an...on Σ"? Since I can't understand that so I don't know what are the p-orbits in the decomposition of Σ ...
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2answers
26 views

Groups, proving inversion to be an isomorphism

True or False? If $G$ is a group, $\psi: G\to G$ is given by $\psi(x)=x^{-1}$, then $\psi$ is an isomorphism.
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2answers
337 views

What is the least $n$ such that it is possible to embed $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?

Let $\operatorname{GL}_2(\mathbb{F}_5)$ be the group of invertible $2\times 2$ matrices over $\mathbb{F}_5$, and $S_n$ be the group of permutations of $n$ objects. What is the least ...
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2answers
67 views

Is orbit a group?

Let a group $G$ act on a set $S$, and let $s$ be an element in $S$. The identity of the orbit of $s$ is $s$ itself and if $a$, $b$ are in the orbit of $s$, then $a b = g_1 g_2 s$, where $g_1$, $g_2$ ...
2
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1answer
40 views

Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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1answer
37 views

Rapid question on minimal normal subgroups

Why does $N$ finite group, $M\unlhd N$ imply that $M$ contains a minimal normal subgroup of $N$? If $M$ is itself minimal in $N$ we have finished. But if not, we know that there exists a minimal ...
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1answer
44 views

How to use group theory to solve larrys square iphone app

There's a 2 d version of rubiks cube on apple app store. How can group theory give an algorithm to solve the iPhone app:larry's square.
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1answer
29 views

Product of two stabilizers of transitive group action is proper subset of G?

Suppose $G$ is a finite group and G acts transitively on some set $X$. Let $a$ and $b$ be two distinct elements of $X$ and $G_{a}$ and $G_{b}$ be stabilizers of $a$ and $b$ respectively.Show that ...
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0answers
23 views

What is the group $_nG$?

I've seen the notation $_nG$, where $G$ is a group and $n > 1$ an integer. What does this notation mean? I suspect it has to do with torsion somehow. Maybe a notation for the $n$-torsion subgroup ...
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1answer
32 views

$R$ is not a direct Sum of its subgroups

How to prove the set of real numbers under addition i.e $(R,+)$ is not the direct sum of two of its proper subgroups?
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votes
3answers
449 views

If G is a group such that all of its proper subgroups are abelian, then G itself must be abelian [closed]

Is statement below true or false? If G is a group such that all of its proper subgroups are abelian, then G itself must be abelian
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0answers
31 views

solvable group and derived series [duplicate]

Can someone prove that group is solvable if and only if its derived series terminate on a trivial group? http://en.wikipedia.org/wiki/Solvable_group
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3answers
38 views

Proving that a set is a group under addition

To show that a set $G$ is a group under addition, do we first need to show that $G$ is closed under addition, or is that implied by proving the three properties of a group, namely there exists an ...
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1answer
37 views

Is it true that group of real numbers under addition isomorphic to group of complex numbers under addition [duplicate]

Is it true that set of all real numbers under addition isomorphic to set of all complex numbers under addition
3
votes
1answer
44 views

Finitely generated group $G$ such that $G\cong G*G$ must be trivial

So, I need to show that a finitely generated group isomorphic to the free product of two copies of itself (obviously thinking every factor as being generated by diferent letters) must be trivial. I ...
1
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1answer
39 views

Endomorphism ring of finite-dimensional representation

If $G$ is any group and $V$ is a finite-dimensional representation of $G$, then we can form the endomorphism ring $E = End_G(V)$. Suppose that $V$ is indecomposable, i.e. not a direct sum of ...
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1answer
55 views

Can you explain more specifically about there is no simple group of size 96

For the solution,I know that the number of Sylow-$2$ subgroups, $n_2=3$, and we can find a subgroup of order $32$,then $G$ can act on the left cosets of $H=P_2$, then it gives a map from $G$ to $S_3$, ...
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1answer
34 views

question on nilpotent group.

Question- If $G$ is a finite group, $N$ a normal nilpotent subgroup of $G$ such that $G/[N,N]$ is nilpotent. Prove that $G$ is nilpotent. How i did it in my exam today- (I know my solution had to be ...
0
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0answers
51 views

Calculations in McLain groups

Is there any program in GAP or other softwares doing calculations in McLain groups and finding subgroups etc. For the McLain groups see Derek Robinson's book: A course in the theory of groups, p347 ...
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1answer
21 views

Basis Composition of permutations

Hi I have the following questions: Calculate the following compositions of permutations on $A=\{0,1,2,3\}$ $(12)(102)$ Ans:$(02)$ $(01)(23)(0123)$ Ans:$(02)$ $(123)(32)$ Ans:$(13)$ ...
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1answer
50 views

Quotient group $G/\{1\}=G$ if $1$ is the identity element of $G$

Is it true that quotient group $G/\{1\}=G$? Or isomorphic to $G$?
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1answer
45 views

Can we define near-rings as some kind of a monoid object in the category of groups?

I recently learned about the tensor product of Abelian groups, which can be used to define the concept 'ring.' In particular, a ring is just a monoid object in the monoidal category of Abelian groups, ...
3
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1answer
55 views

Hecke Operators

I'm currently working my way through the section on Hecke operators in Serre's book. In the proof that $ T(m)T(n) = T(mn) $ for all $ m,n \in \mathbb{Z}_{\geq 1} $ with $ \textit{gcd}(m,n) = 1 $. In ...
3
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0answers
45 views

Why are centers, centralizers and normalizers called that way?

I know what they are, but where do the names come from?
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1answer
37 views

Upper bound for the number of generators of a group

Let $H\leq G$ and $x\in G$. If $H$ is generated by at most $n$ elements, prove that $\langle H,x\rangle$ is generated by at most $n+1$ elements. This is intuitively obvious but when I try to ...
2
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1answer
52 views

Solvable groups in group theory

If $N \unlhd G$, and $M,K \leq G$ such that $M \unlhd K$, then does it imply that $MN \unlhd KN$? If yes, how?
3
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1answer
54 views

Direct product of two groups.

Let $N$ be a minimal normal subgroup of $G$. Also let $N$ and $\frac{G}{N}$ are non-abelian simple. Can we say that $G=N\times A$ where $A$ is a non-abelian simple subgroup of $G$?
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1answer
49 views

$H, K$ are subgroup of $G$. If $H \cup K \leq G$, then $H \subseteq K$ or $K \subseteq H$?

Is the statement True or False? Let $H,K$ be subgroups of a group $G$. If $H \cup K \leq G$, then $H \subseteq K$ or $K \subseteq H$. Need help with this question.
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1answer
42 views

non-abelian simple group

Let G be a non-solvable group and $(\frac{G}{Z(G)})^{'}=\frac{G}{Z(G)}$. Can we say that there is a normal subgroup $N$ of $G$ with property $Z(G)\leq N$ such that $\frac{G}{N}$ is a non-abelian ...
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2answers
40 views

Let $ G$ be a nilpotent group. prove that there exist $a\in G$, such that $ o(a)=exp(G)$.

Please hint me. I want to proof the following homework: Let $ G$ be a nilpotent group. prove that there exist $a\in G$, such that $ o(a)=exp(G)$.
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3answers
84 views

Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?

Is the following statement true or false: If $G$ is a group with the property that $g=g^{-1}$ for all $g \in G$, then $G$ is abelian. I believe it is false since I know that abelian ...
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1answer
39 views

fibred product of groups of multiplicative type (or, more generally, of linear algebraic groups)

Let $M, M', M''$ be $k$-groups of multiplicative type, and let $M' \to M$ and $M'' \to M$ be morphisms of group schemes. Is the fibred product $M' \times_M M''$ a $k$-group of multiplicative ...
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3answers
60 views

If $N$ normal subgroup of $G$ and $M$ normal subgroup of $G$ prove $MN$ is a subgroup [closed]

If $N\vartriangleleft G$ and $M\vartriangleleft G$ and $MN = \{mn | m \in M, n \in N\}$, prove that $MN$ is a subgroup of $G$ and that $MN\vartriangleleft G$
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1answer
63 views

Collection of $n - 1$ bijections on the set of $n$ elements

Let $\sigma_1, \ldots \sigma_{n-1} \in S_n \setminus \{\operatorname{id}\}$. Does there exists $k \in \{1,2, \ldots n\}$ such that $k \sigma_1 \ne k \sigma_2 \ne \cdots \ne k \sigma_{n-1}$?
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1answer
23 views

Abstract Monomorphism 3 part Question

I have been working on this problem for an hour now and gotten nowhere: Let $G$ be any group and $A(G)$ the set of all 1-1 mappings of $G$, as a set, onto itself. Define $L_a : G \rightarrow G$ by ...
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1answer
43 views

If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$

If $G$ is a finite group and its $p$-sylow subgroup $P$ lies in the center of $G$, prove that there exists a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$
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0answers
29 views

The additive reals and the multiplicative positive reals

I get the concept that there is a relationship between the additive reals and the positive reals with multiplication, but more generally, how come a function like $e^{x+y}=e^x*e^y$ is a suitable ...
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3answers
84 views

Is there a simple group of order $105$?

By Sylow theorem, we see the number of the $7$-sylow subgroup is $n_7$. Then $n_7=1$ (mod $7$) and $n_7$ divides $15$; thus $n_7=15$, but why do we have $6\cdot 15=90$ elements of order $7$? And just ...
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0answers
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How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up ...
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1answer
23 views

Correctness of proof that the commutative operation * on a binary structure is a structural property [duplicate]

Here is what I have as a proof for now. Can you tell me where I need to edit it and or how it should be instead? Let and be two arbitrary binary structures with an isomorphism f: S->T. Assume * is ...
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2answers
51 views

what is the order of the subgroup of $S_3$

True or false? $S_{3}$ has a subgroup of order $5$. $S_3$ is the group of all permutations of the numbers $1,2,$ and $3$. An example of a permutation is, for instance, $(32)$, which means switching ...
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0answers
29 views

Composition series in an alternating group

I have seen that a composition series for $A_4$ is $$1<\{1, (1 2)(3 4)\}<\{1,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)\}<A_4$$ But how can I find it? I mean when I am asked to find a composition ...
2
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1answer
38 views

prove that the no of normal subgroups of a fixed order containing K is congruent to 1 mod p.

Let G be a p group K is a normal subgroup of G of order p to the power a. Then prove that the no of normal subgroups of a fixed order containing K is congruent to 1 mod p. I've proved that the no of ...
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1answer
16 views

How to show that any solvable transitive subgroup of S$_p$ where $p$ is a prime has a conjugate contained in Aff($\mathbf F_p$)?

Here Aff ($\mathbf F_p$) denotes the group of affine transformations $x\rightarrow ax+b,$ with $ a\neq 0, b\in \mathbf F_p$. What I've done is to show that the penultimate group in the solvable series ...
1
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1answer
42 views

Prove that an associative operation* is a structural property

First off how does one prove that an operation $\ast$ is a structural property and how is this different from just proving an isomorphism? My main question is: how can I prove that $\ast$ on a binary ...
0
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0answers
46 views

Nilpotent Groups

Is there any easier way to prove that subgroups and quotients of a nilpotent group(finite or infinite) is nilpotent. This is one of the exercises in Hungerfords Algebra book(Pg 106,#5).How to use thm ...