Homework SourseShow that the diagonal x xx G is a subgroup of G G and this
Show that the diagonal {(x, x)|x G} is a subgroup of G × G and this subgroup is isomorphic to G.
Show that the diagonal {(x, x)|x G} is a subgroup of G × G and this subgroup is isomorphic to G.
Show that the diagonal {(x, x)|x G} is a subgroup of G × G and this subgroup is isomorphic to G.
Solution
solution-: Let G = {r R | r is not = 1}. We define an operation on G by a b = a + b + ab
for all a, b G. Note that G is a group.
Define a map
\' : G R by
\'(a) = 1 + a
Since a is not = 1 means that 1 + a 6= 0, it is clear that \' is a bijective map from G to R.
Furthermore, we have
\'(a b) = \'(a + b + ab) = 1 + a + b + ab = (1 + a)(1 + b) = \'(a)\'(b) .
Hence \' is a homomorphism. Therefore, \' is indeed an isomorphism from G to R and so
those two groups are indeed isomorphic.
