#(ZZ_5, o+_5)# is the set of integers modulo #5# with addition modulo #5#.
That is "clock addition" modulo #5#.
Since #5# is prime, there are only two subsets which are closed under addition, namely the whole set #{0, 1, 2, 3, 4}# and the set containing only #{0}#.
In other words, the only two subgroups are:
#(ZZ_5, o+_5)# and #({0}, +)#
Here's the addition table for the group #(ZZ_5, o+_5)#:
#underline(color(white)(0)o+_5|color(white)(0)0color(white)(00)1color(white)(00)2color(white)(00)3color(white)(00)4color(white)(0))#
#color(white)(o+_5)0|color(white)(0)0color(white)(00)1color(white)(00)2color(white)(00)3color(white)(00)4#
#color(white)(o+_5)1|color(white)(0)1color(white)(00)2color(white)(00)3color(white)(00)4color(white)(00)0#
#color(white)(o+_5)2|color(white)(0)2color(white)(00)3color(white)(00)4color(white)(00)0color(white)(00)1#
#color(white)(o+_5)3|color(white)(0)3color(white)(00)4color(white)(00)0color(white)(00)1color(white)(00)2#
#color(white)(o+_5)4|color(white)(0)4color(white)(00)0color(white)(00)1color(white)(00)2color(white)(00)3#
