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

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Order of a permutation

What does the order of a permutation actually mean? I accept the fact that it is the l.c.m. of the lengths of the cycles in its cycle decomposition, but I don't really have an intuition for what the ...
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0answers
22 views

Realizete groups as a unit group of ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
1
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2answers
19 views

Constructing group homomorphisms.

Let $G$ and $H$ be groups, and suppose I want to construct a group homomorphism $\phi$ between them. From what I know, I just need to send each element $x \in G$ to an element $y \in H$ such that the ...
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0answers
27 views

Property of group G with $|G|=2n$ with $n$ elements of order $2$ (Sylow theorem application)

Suppose $G$ is a group such that $|G|=2n$, $G$ has $n$ elements of order $2$ and the rest of the elements form a subgroup $H$. Show that $H \lhd G$ and $n$ is odd. I am pretty lost with this ...
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1answer
16 views

What is an example of a proper normal subgroup of the kernel of a homomorphism?

I'm reading this proof by Hungerford that concerns any normal subgroup of the kernel of a homomorphism. I understand the proof well enough, but I wanted to have some concrete example to guide or ...
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0answers
23 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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0answers
15 views

Prove that the normaliser of a Sylow $2$-subgroups of $D_{2n}$ is itself.

Let $2n = 2^a k$ for $k$ odd. Prove that the normaliser of a Sylow $2$-subgroups of $D_{2n}$ is itself. Here |N|||G| where N is the normaliser of a sylow 2 subgroup H. But N can properly contain H. ...
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0answers
23 views

Self normalising p sylow

When are p-sylow subgroups self normalising? I know, for example, that if the group has order $ p^2q^2$ then the p-sylow subgroups are self-normalising if there are $q^2$ of them. I just don't know ...
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0answers
26 views

Equivalence of Nilpotence in group

We have 2 definition of Nilpotent group.I am trying to prove the equivalence. A central series for G is a normal series 1=G0⊴G1⊴...⊴Gr=G such that Gi/Gi−1=Z(G/Gi−1) for every i=1,...,r. An arbitrary ...
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1answer
40 views

Determining if a set is a group

Let $S=\lbrace x+y\sqrt2 : x,y\in \mathbb R \rbrace$ \ $\lbrace0\rbrace$. Justify whether $S$, together with traditional multiplication, is a group. I've verified that the set is closed under the ...
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3answers
22 views

Prove Equivalence Relation in G

Hei, guys! I'm having some trouble with the next problem: Let $A$ and $B$ be subgroups of $G$. Show that $\sim$ is an equivalence relation when it is defined as follows: $g\sim g'\Leftrightarrow g' ...
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0answers
27 views

$G$ a finite group $n$-abelian goup and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then to show $G$ is abelian [duplicate]

Let $G$ be a finite group and $n$ be a given positive integer such that $(ab)^n=a^nb^n , \forall a,b \in G$ and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then how to prove that $G$ is abelian ? If I can show ...
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2answers
44 views

When can an infinite abelian group be embedded in a field?

This question comes from this question by user72870. That question would easily be answered if we know the cyclicity of the group in question, but, as the OP appears to be trying to prove that the ...
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0answers
35 views

Isomorphism of a set [duplicate]

We know that $\operatorname {Aut}(G) \over \operatorname {Inn}(G)$ $\cong \operatorname {Out}(G)$. Is it true that $\operatorname {Aut}(G) \cong \operatorname {Inn}(G) \rtimes \operatorname {Out}(G)? ...
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1answer
44 views

automorphism group of groups [on hold]

Given a group $G$, I would like to calculate $\operatorname{Aut}(G)$. From definition of $\operatorname{Aut}()$ we know: $\operatorname{Aut}(G)\le \operatorname{Sym}(G) $ If the group is finitely ...
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0answers
23 views

prove that $(E_{p^n},*)$ is cyclic group

if $p \in$ $\mathbb{N}$ is a prime integer, how can i prove that $E_{p^n}$ the group of invertible elements of $\frac{\mathbb{Z}}{p^n\mathbb{Z}}$ is a cyclic group.
2
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1answer
38 views

Transitive action on two sets! [duplicate]

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of ...
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0answers
34 views

Why the paired orbit has the same size here?

enter link description here On this proof, he just showed that the the paired orbit of (a,b) has the same size, but this seem has nothing to do with the size of corresponding paired orbit of an orbit ...
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0answers
35 views

Intuition fail with normal subgroups

My intuition fails me, when I try to undestand normal subgroups. I read that Alternating group $A_n$ is simple for all $n \geq 5$, $n$ is the order of the group. So $A_4$ is possibly (this has to do ...
2
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0answers
17 views

Sufficiently transitive implies alternating sans Enormous Theorem

According to this webpage and this mathworld article, if $G<S_n$ is a permutation group which acts sextuply transitively then $G=A_n$ is the alternating group, but this fact is known on the basis ...
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1answer
66 views

Some functorial maps $G\times G\rightarrow G$

Let $G$ be a group. Le diagonal map $\delta:G\rightarrow G\times G$ obviously gives a functorial morphism from the identity functor of $\mathbf{Grp}$ to the functor $P$ sending $G\mapsto G\times G$ ...
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2answers
39 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
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0answers
15 views

Is $(Nb)^{o(N)}=e$, where $N$ is a normal subgroup of group $G$?

Let $N$ be a normal subgroup of group $G$. Let $Nb\in G/H$ be a coset, where $b\notin N$. Herstein says that $(Nb)^{o(N)}=e$, where $o(N)$ is the order of the subgroup $N$. I don't understand why. ...
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0answers
21 views

Crossed homomorphism: $\varphi(x)=\varphi(y)\iff Kx=Ky$

Let $G$ be a finite group, $N\unlhd G$. A crossed homomorphism from $G$ to $N$ is defined as a $\varphi:G\to N$ s.t. $\varphi(xy)=\varphi(x)^y\varphi(y)$. It's not in general a group homomorphism. ...
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1answer
19 views

How to show $F(S)$ is normal in Sym$(S)$

As a follow-up of this question On non-trivial normal subgroup(s) of $A(S)$ , where $S$ is infinite , how do we show that the finitary symmetric group $F(S)$ is a subgroup and normal in Sym $(S)$ , ...
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0answers
21 views

Determining all homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ? [duplicate]

What are all homomorphisms from $\mathbb Z_n$ to $\mathbb Z_n$ ? I know about all automorphisms but am not clear about all homomorphisms ; are there a total of $n$ homomorphisms ? In general , what ...
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0answers
66 views

Is $\text{Aut}(A \times B) = \text{Aut}(A) \times \text{Aut}(B)$? [on hold]

Is $\text{Aut}(A \times B) = \text{Aut}(A) \times \text{Aut}(B))$? where $A$ and $B$ are subgroups, $A \times B$ is the direct product and $\text{Aut}(A)$ refers to group of all automorphisms of ...
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1answer
18 views

If $\sigma \in Aut(G)$where $|G|$ odd has order 2 there exists Sylow $p$-subgroup with $\sigma(P)=P$

Let $G$ be a group of odd order and $\sigma$ an automorphism of G of order 2. Show that if the prime $p$ divides $|G|$ then there exist a Sylow $p$-subgroup $P$ such that $\sigma(P)=P$.
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1answer
40 views

How to show that the orbits of the action of Gs on S \ {s} have lengths that are equal in pairs.

Question:Let G be a group of odd order acting transitively on a set S. Fix s ∈ S. Show that the orbits of the action of Gs on S \ {s} have lengths that are equal in pairs. My idea: set a point ...
4
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1answer
18 views

What do this quotient group represent

If $C^*$ and $R^+$ denote the multiplicative group of non zero complex numbers and the subgroup of positive reals, then what does the quotient group $\frac{C^*}{R^+}$ mean?
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1answer
62 views

Automorphisms of the group of integers $\mathbb Z$

Can anyone help me showing $\operatorname{Aut}(\mathbb Z)\simeq \mathbb Z_2$? I guess I should define an homomorfism $\phi:\mathbb Z\longrightarrow S(\mathbb Z)$ with kernel $2\mathbb Z$ and image ...
2
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1answer
40 views

Order of $a^m$.

Let $G = \langle a \rangle$ a finite cyclic group of order $n$. Prove that $|a^m| = \frac{n}{\gcd(m,n)} = \frac{\mathrm{lcm} (m,n)}{m}$. I managed "half" of it. Write $|a^m| = k$ and $d = ...
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1answer
74 views

How many automorphisms does $S_3\times S_3$ have?

I've shown that $|\text{Aut}(S_3\times S_3)|\ge 72$, how can I show that $|\text{Aut}(S_3\times S_3)|\le 72$ ?
3
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2answers
55 views

Does $G=HK_1=HK_2$ imply $K_1=K_2$ in a locally cyclic group?

Let $G$ be a locally cyclic group and $H,K_1,K_2\le G$ be subgroups such that $G=HK_1=HK_2$ and $H\cap K_1=H\cap K_2=\{1\}$. Is $K_1=K_2$?
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0answers
33 views

For $n>4$ the only nontrivial subgroup of symmetric group of order $n$ is alternating group. [duplicate]

For $n>4$ the only nontrivial subgroup of symmetric group of order $n$ is alternating group. If I use that $A_n$ is simple (although I don't know the proof ) Next what do I do?
7
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1answer
114 views

If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian

Let $G$ be a group. If $(ab)^n=a^nb^n$ $\forall a,b \in G$ and $(|G|, n(n-1))=1$ then prove that $G$ is abelian. What I have proved that If $G$ is a group such that $(ab)^i = a^ib^i$ for three ...
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1answer
37 views

Let $H\le G$ s. t. whenever $Ha≠Hb$ then $aH≠bH$. Prove that $gHg^{−1}\le H\;$ $\forall g\in G$. [on hold]

Suppose that H is a subgroup of G such that whenever$ Ha \ne Hb $ then $ aH \ne bH $. Prove that $ gHg^{-1} \subseteq H$ for all g in G.
2
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1answer
32 views

Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
3
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2answers
28 views

If a subgroup acts transitively on a set, then the index of the subgroup equals the index of the stabilizer?

I am trying to prove the following: If a subgroup $H < G$ acts transitively on a set $X$, then $[G:H] = [G_x:H_x]$ for any $x \in X$ ($H_x$ denotes the point stabilizer of $x$ in $H$.) Any hints ...
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2answers
32 views

Orbit and stabaliser of $2\times2 * 2\times1$ matrices

I have the group action of matrix multiplication, meaning: $g((x,y))=\begin{pmatrix}a&0\\0&b\end{pmatrix}$$ \begin{pmatrix}x\\ y\end{pmatrix}$ ...
2
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0answers
54 views

If $X$ generates $\Bbb{Q}$ then $X\setminus\{x\}$ also generates $\Bbb{Q}$ [duplicate]

If $X$ is a generator subset of $\Bbb{Q}$ then for $x\in X$, $X\setminus\{x\}$ also generates $\Bbb{Q}$. Clearly if I can express $x$ as a combination of the remaining generators we are done. ...
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1answer
31 views

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
40 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
50 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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0answers
30 views

Cyclic groups in $S_4$ with proofs [on hold]

True or false? Every element of $S_4$ is a cycle. The answer is false. But how to prove is the difficulity
1
vote
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.
22
votes
2answers
286 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 ...
1
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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 $g_1$, $g_2$ ...