Group Theory Permutation (Hints and Help Please)

  • Thread starter MellyVG257
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Homework Statement



1. Let n ≥ 2. Let H = {σ ∈ S_n: ord(σ) = 2}. Decide whether or not H is a subgroup of S_n.
2. Let G be a group of even order. Show that the cardinality of the set of elements of G that have order 2 is odd.

The Attempt at a Solution



1. I have no idea where to start with this. I tried looking at the rule and using the 3 rules for H to be a subgroup to prove it, but I don't know what to do with the order. Really lost with this. Any ideas on how to start it?

2. I tried to solve it using order as being the cardinality of the set. But I think I am missing something. I know what a group with order 2 is [(12)(24), right?] and I know that the sgn for even and odd and 1, -1 respectively. But this is where I became stuck. Any ideas?
 

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