Tagged Questions

209 views

Is there a group-theoretic proof of the Riemann rearrangement theorem?

The analytic proofs of the Riemann rearrangement theorem are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I ...
45 views

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? [on hold]

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? Does there exist a subgroup of $G$ of order $180$?
41 views

67 views

60 views

How did the Symmetric group and Alternating group come to be named as such?

The Dihedral group makes sense, "Di" means two, and "hedral" means.. shape I think (I've just realised how much of what I think words mean are guesses based on experience) like a "polygon" is a 2d ...
57 views

How can I prove that $C_{S_4}((12)(34))$ is a subgroup of order 8?

I would like to know if there is a smart method (i.e. avoid to make all by hand) to understand the order of $C_{S_4}((12)(34))$. The only thing I thought is that ...
70 views

Prove $S_4$ has only 1 subgroup of order 12

The subgroup in $S_4$ that I know has order 12 is the subgroup of all even permutations, otherwise known as the alternating group $A_4$. However, I know this from a fact and not because I am able to ...
37 views

Subgroup of $S_4$ with order 12 [duplicate]

I need to find a subgroup of $S_4$ that has order 12 other than the subgroup of even permutations or anything isomorphic to it. What is the procedure to go about finding such a subgroup?
42 views

Proof that $S_3$ is the smallest non-commutative group

Before I start attempting to show that every group of order less than 6 is commutative, is there a shorter/faster way to go about proving this?
52 views

Normalizer of a Sylow subgroup

In this question Sylow subgroups of soluble groups Jack Schmidt mentions that the normalizer of $P$ in $S_p$ is solvable. Suppose $P$ is generated by a cycle of length $p$. Could you provide any ...
84 views

Embedding $S_n$ into $A_{n+2}$

I am trying to prove that for all $n$, $S_n$ is isomorphic to a subgroup of $A_{n+2}$. Say $S_n$ acts on $\{\alpha_1,...,\alpha_n\}$ and $A_{n+2}$ acts on ...
84 views

Generating the symmetric group $S_n$

I know that $\sigma =(1 2 \ldots n)$ and $\tau =(1 2)$ together should generate the symmetric group by virtue of conjugation, i.e. $(\sigma)^k \circ \tau \circ (\sigma^{-1})^k = (k+1, k+2)$; we know ...
67 views

Standard represention of $S_3$

I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in the ...
79 views

What does permutation stand for as a power?

I am just reading some books about abstract algebra and I don't understand what a permutation stands for as a power. For example, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
57 views

59 views

Find homomorphisms from $C_8$ to $S_3$ and from $C_6$ to $S_3$, and are there any isomorphisms?

For $C_8$ to $S_3$: Let $C_8$ be generated by $x$. Then $C_8 = \langle x : x^8 = e \rangle = \{e,x,x^2,x^3,x^4,x^5,x^6,x^7\}$. But how do I show homormorphisms? I'm supposed to pick elements of $x$ ...
Sylow $p$ subgroups of $S_{2p}$ and $S_{p^2}$ - Dummit foote - $4.5.45; 4.5.46$
Question is to find Sylow $p$ subgroups of $S_{2p}$ for odd prime $p$ and show that this is an abelian group of order $p^2$ Sylow $p$ subgroups of $S_{p^2}$ for odd prime $p$ and show that this is ...