abstract algebra

If G is any group, define $:G->G by $(g) = g^-1. Show that G is abelian if an only if $ is a homomorphism.

abstract algebra

If $:G->G1 is a homomorphism, show that K = the set of g belonging to G given that $(g)=1 is a subgroup of G (called the kernel of $) Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

Note: ~~ means an isomorphism exists. Moreover,if an isomorphism existed from G to G1 I would say G ~~ G1 Questions: If G is an infinite cyclic group, show that G ~~ Z (Z is the set of integers) Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

If G = and $:G->G1 is an onto homomorphism, show that G1 = <$(X)>, where $(X) = the set of $(x) given that x belongs to X. Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

If H is a subgroup of G, define a mapping $ from the right cosets of H to the left cosets by $(Ha) = a^-1H. Show that $ is a (well defined) bijection. Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.