abstract algebra solve ONLY A6 PLEASE AND THANK YOU Recogni

abstract algebra
solve ONLY
A6!
PLEASE AND THANK YOU

. Recognizing Subgroups In parts 1-6 below, determene whether or not Ff is a subgroup of G. (Assume tht the operation of H is the same as that of G.) Instructions tfH is a subpop of G, show that both conditions in the definition of \"Subgroup\" are tion of\" subgroup\" are satisfied I H is nor a subgroup ofG, explain which condition Eaals Example G-R, he mmutaplicaive group of the rea whers H 2\' . \"Ezl His is not asubgroup ofG.

Solution

We know that a group is an algebraic structure comprising a set of elements and equipped with an operation that combines any two elements to form a third element. The operation has to satisfy the four conditions (called the group axioms) of closure, associativity, identity and invertibility.

Here, H is a subset of G . We have to ascertain, whether H is a subgroup of < RxR, +>. We know that (0 , 0) is the additive identity of R XR . However, since 0² + 0² = 0 is not greater than 0, hence (0, 0) does not belong to H. Therefore, H is not a subgroup of < RxR, +>.