Finding all sub groups

Question

Can you please help me find all the sub groups of Z₄*Z₈*Z₁₆  of order 4

Answer 1

Hi Rebecca! The subgroups of a direct product group are just going to be all combinations of subgroups of each group. In other words, the subgroups will be of the form: {subgroup of Z₄}, {subgroup of Z₈},

{subgroup of Z₁₆}. For example, the group Z₂*Z₂*Z₂ is a subgroup of Z₄*Z₈*Z₁₆, since Z₂ is a subgroup Z_4,Z_8, and Z₁₆. Can you check and verify that any group of this form will be a subgroup?

Now we have to find the set of subgroups that have four elements. We can find the order of a direct product group by multiplying the order of each constituent group (ie |Z₄|*|Z₈|*|Z₁₆|). Thus we just need to look for all possible subgroups of Z_4,Z_8, and Z₁₆ whose order multiplies to 4.

I hope this helps -- please let me know if you have any questions!