I've somewhat recently been going back through one of my brother's old textbooks reviewing group theory. I'm up to a chapter called Factor-Group Computations and Simple Groups. The problems at the end seem to have me stumped and I want to make sure I'm understanding enough before I proceed.
There are a dozen problems asking to classify a given group according to the fundamental theorem of finitely generated abelian groups. These are a couple that stumped me.
The first is $(Z_4\times Z_4\times Z_8)/\langle (1,2,4)\rangle$. I can see that $Z_4\times Z_4\times Z_8$ has order $128$ while $\langle(1,2,4)\rangle$ has order $4$, so the factor group in question has order $32$. It also appears there are elements of up to order $8$, which I think should narrow it down to $Z_4\times Z_8$ or $Z_2\times Z_2\times Z_8$, but how do I tell which?
This next one I have no clue how to proceed with. $(Z\times Z)/\langle(1,2)\rangle$ How would I go about solving this one?
Best Answer
A hint for your first question: Subtracting a multiple of $(1,2,4)$ from any element $(a,b,c)$ of $\mathbf{Z}_4\times\mathbf{Z}_4\times\mathbf{Z}_8$ allows us to make the first coordinate equal to zero. So within each coset of $\langle(1,2,4)\rangle$ there is an element with first coordinate equal to zero. Why does this imply that the quotient group can be generated by at most two elements?
A hint for your second question: Can you think of a basis of $\mathbf{Z}\times\mathbf{Z}$ containing the element $(1,2)$?