Subgroup of external direct product

[xmcestmoi]

I am trying to do the followin 2 problems but not sure if I am doing them correct.
Anyone please have a look...


1. In Z40⊕Z30, find two subgroups of order 12.

since 12 is the least common multiple of 4 and 3, and 12 is also least common multiple of 4 and 6.
take 10 in Z40, and 10 generates a subgroup of Z40 of order 4, that is <10>={0,10,20,30}

take 10 in Z30, 10 generates a subgroup of Z30 of order 3, that is <10>={0,10,20}

take 5 in Z30, 5 generates a subgroup of Z30 of order 6, that is <5>={0,5,10,15,20,25}

Answer: two subgroups of Z40+Z30 with order 12 are <(10,10)> and <(10, 5)>


2. Find a subgroup of Z12⊕Z18 isomorphic to Z9⊕Z4.

order of Z9⊕Z4 is 36, which is the least common multiple of 9 and 4.

Now find a subgroup of Z12⊕Z18 with order 36.
take 3 in Z12, 3 generates a subgroup of Z12 with order 4, that is <3>={0,3,6,9}

take 2 in Z18, then 2 generates a subgroup of Z18 with order 9, that is <2>={0,2,4,6,8,10,12,14,16}.

Answer: a subgroup isomorphic to Z9+Z4 is <(3,2)> in Z12⊕Z18.

 

[Mathnerdmo]

(1) looks good.

In (2), your answer is correct, but all you've shown is a subgroup with the same order as Z9⊕Z4. Being isomorphic is much stronger than having the same order, though, so you're not finished on (2) yet. Try to exhibit an isomorphism, e.g.

 

[xmcestmoi]

Thank you :) I will try to come up with an isomorphism from <(3,2)> to Z9⊕Z4