# Order of abelian groups

• June 20th 2008, 01:50 PM
kleenex
Order of abelian groups
I have difficulty to do the following question,

(a)Determine (up to isomorphism) all abelian groups of order 120.
$C_2 \times C_2 \times C_2 \times C_3 \times C_5$
$C_2 \times C_4 \times C_3 \times C_5$
$C_8 \times C_3 \times C_5$

(b)Verify that $Z_{20} \times Z_{24}/<(5,6)>$ is an abelian group of order 120. To which of the groups listed in part (a) is isomorphic?

Part (a) should be correct but I don't know how to do (b). Please help me, thank you.
• June 22nd 2008, 07:15 AM
ThePerfectHacker
Quote:

Originally Posted by kleenex
(b)Verify that $Z_{20} \times Z_{24}/<(5,6)>$ is an abelian group of order 120. To which of the groups listed in part (a) is isomorphic?

Hint: $(G_1\times G_2)/(H_1\times H_2) \simeq (G_1/H_1)\times (G_2/H_2)$.
Where $H_1,H_2$ are normal subgroups.