... discussion of graded k-algebras, for k a commutative ring, leads totensor algebras, central simple algebras and the Brauer group, exterior algebra (includingGrassmann algebras and the binomial ... then their direct product, denoted by H × K,istheset of all ordered pairs (h, k) with h ∈ H and k ∈ K equipped with the operation(h, k)(h, k) = (hh, kk).It is easy to check that ... surjective homomorphism with ker f = K × K.Thefirst isomorphism theorem nowgives the desired isomorphism. •Proposition 2.80. If G is a group containing normal subgroups H and K with H ∩ K ={1}...