Isomorphism types of abelian groups

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The discussion focuses on identifying the isomorphism types of abelian groups for the orders 74 and 800. For order 74, the groups are Z(74) isomorphic to Z2 * Z37, resulting in two distinct groups: Z74 and Z2 * Z37. For order 800, which factors into 2^5 * 5^2, the groups include Z25, Z5 x Z5, and six distinct groups from the 2^5 component, leading to a total of twelve unique abelian groups: Z25 x Z32, Z25 x Z16 x Z2, Z25 x Z8 x Z4, Z25 x Z4 x Z4 x Z2, Z25 x Z4 x Z2 x Z2 x Z2, Z25 x Z2 x Z2 x Z2 x Z2 x Z2, Z5 x Z5 x Z32, Z5 x Z5 x Z16 x Z2, Z5 x Z5 x Z8 x Z4, Z5 x Z5 x Z4 x Z4 x Z2, Z5 x Z5 x Z4 x Z2 x Z2 x Z2, and Z5 x Z5 x Z2 x Z2 x Z2 x Z2 x Z2.

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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?
 
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What does the fundamental theorem say? What have you tried so far?
 
Number Nine said:
What does the fundamental theorem say? What have you tried so far?

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
 

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