Isomorphism types of abelian groups

1. Nov 25, 2012

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??????

2. Nov 25, 2012

Number Nine

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

3. Nov 25, 2012

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