Well, in 5 years of PF'ing and watching over this forum, I am finally posting my first homework question. :tongue2: I'm taking a graduate course in Algebra, and it's been 11 years since I took the undergraduate version. So, I'm going back and doing all the homework exercises in my undergrad book. I'm stuck on this one.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

[itex]S_A[/itex] is the group of all permutations of a set [itex]A[/itex] under permutation multiplication. [itex]B[/itex] is a subset of [itex]A[/itex], and [itex]b[/itex] is a particular element of [itex]B[/itex]. Determine whether the given set is sure to be a subgroup of [itex]S_A[/itex] under the induced operation. Here [itex]\sigma=\{\sigma(x)|x \in B\}[/itex]

And the subsets are...

[itex]H=\{\sigma\in S_A|\sigma\subseteq B\}[/itex]

[itex]K=\{\sigma\in S_A|\sigma=B\}[/itex]

2. Relevant equations

Not applicable.

3. The attempt at a solution

First let's consider [itex]H[/itex]. The elements of [itex]H[/itex] are all of the permutations that send the elements of [itex]B[/itex] to a subset of [itex]B[/itex]. To try to grasp this, I considered an example.

Let [itex]A=\{1,2,3,4,5\}[/itex] and [itex]B=\{1,2,3\}[/itex]. Then choose a permutation [itex]\sigma_1[/itex] that satisfies the condition of membership in [itex]H[/itex].

[tex]\sigma_1=\left(\begin{array}{ccccc}1 & 2 & 3 & 4 & 5\\3 & 1 & 2 & 5 & 4 \end{array}\right)[/tex]

When I look at this, I can't see how [itex]H[/itex] could be anything other than [itex]K[/itex] itself. If the image of [itex]B[/itex] under [itex]\sigma_1[/itex] is anything other than [itex]B[/itex], then it contains elements of [itex]A[/itex] that donotbelong to [itex]B[/itex]. Hence, the image would not be a subset of [itex]B[/itex].

The answer in the back of the book says that [itex]H[/itex] is not a subgroup of [itex]S_A[/itex], as it is not closed under taking of inverses. I do not see how that could possibly be right.

I'll leave [itex]K[/itex] alone until I get [itex]H[/itex] sorted out.

Thanks,

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Subset of the Group of Permutations: Subgroup or Not?

**Physics Forums | Science Articles, Homework Help, Discussion**