# Relation between two subgroups and the group with combined generators

Consider a group $G$, with subgroup $A$ and $B$ which have generators $\langle A_1,\ldots,A_m\rangle$ and $\langle B_1,\ldots,B_n\rangle$ respectively. How is the group $C$ with generators $\langle A_1,\ldots,A_m, B_1,\ldots,B_n\rangle$ related to $A$ and $B$? Is it simply the case that $C=A\times B$ or something different - either way please can you explain?

Consider the free group $G$ with generators $a$, $b$. This is a non-commutative group with elements the reduced words in $a$, $b$, $a^{-1}$ and $b^{-1}$, that is concatenations of these "letters" with no letter adjacent to its inverse. But $A=\langle a\rangle$ and $B=\langle b\rangle$ are each infinite cyclic (so Abelian groups). Then $C=\langle a,b\rangle =G$ is not isomorphic to $A\times B$.
If A and B don't sit in some larger group C, then you haven't defined a multiplication between elements of $A$ and $B$ yet so the question doesn't make sense in that case. In the example Lord Shark gave, there he defined a multiplication between the two groups, namely concatenation.
• I have accounted for this by saying that $A$ and $B$ are subgroups of $G$. – Quantum spaghettification Jun 3 '18 at 7:27
• My bad, missed that. In that case, your intuition should definitely not point to $A\times B$. A direct product is when you want to two groups $A,B$ to not interact at all. Since you have a multiplication already defined between the two, <A,B> will be some new group entirely. – Paul T Jun 3 '18 at 7:35