Group, Help

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superkillball

Would any do question 1(vi) and 3(ii) and (iv)?
Thank you

1vi) is a group. In fact it generally is for any power set P(S) of subsets of S. The identity is the emptyset and the inverse of A is A. Leave associativity to you.

3ii) H_1 isn't a subgroup as what is the inverse of 1. H_2 is a subgroup - check the subgroup axioms.

3iv) H_1 is (recall that composition of maps is always associative), but H_2 isn't as the identity map wouldn't be in there.

Reply 2

J.F.N

3ii. H1 is not a subgroup since the additive inverses do not exist. H2 is a subgroup since it is closed, associative, every element has an inverse, and the identity 0 exists.
3iv. H1 is a subgroup, H2 is not a subgroup since the identity does not exist (the identity being f(x)=x, which is clearly impossible if f(2)=0).