isomorphism types of abelian groups

cummings12332

wrtie down the possible isomorphism types of abelian groups of orders 74 and 800

then for 74=2*37 then Z(74) is isomorphism to Z2 * Z37 (by chinese remainder theorem) then for 74 , 2 we have Z74 and Z2*Z37 (i am not sure it is right or wrong
then for 800 i know i should apply the fundamental theorem of fintely generated abelian groups, but i am not quite understand the methods of it , can someone help me the 800 case??????

Number Nine

What does the fundamental theorem say? What have you tried so far?

cummings12332

for 800=2^5*5^2 then there are two abelian groups of order 25:Z25,Z5xZ5, for2^5 ,we have 6 abelian groups : Z32,Z16xZ2,Z8xZ4,Z4xZ4xZ2,Z4xZ2xZ2xZ2,Z2xZ2xZ2xZ2xZ2

so we get 12 different groups:
Z25xZ32
Z25xZ16xZ2
Z25xZ8xZ4
Z25xZ4xZ4xZ2
Z25xZ4xZ2xZ2xZ2
Z25xZ2xZ2xZ2xZ2xZ2
Z5xZ5xZ32
Z5xZ5xZ16xZ2
Z5xZ5xZ8xZ4
Z5xZ5xZ4xZ4xZ2
Z5xZ5xZ4xZ2xZ2xZ2
Z5xZ5xZ2xZ2xZ2xZ2xZ2