# Given two isomorphic subgroups of a finite abelian group with the same number of minimal generators, are the quotients isomorphic?

Let $$G$$ be a finite abelian group.

We know that the least size of a minimal generating set of a proper nontrivial subgroup $$H \leqslant G$$ might be the same as $$G$$, for example ($$\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2$$ has a subgroup isomorphic to $$\Bbb Z_2$$, and both $$\Bbb Z_6$$ and $$\Bbb Z_2$$ can be generated by $$1$$ element). Also, in general, given two isomorphic subgroups $$H_1$$ and $$H_2$$ of $$G$$, it is not necessarily true that $$G/H_1$$ and $$G/H_2$$ are isomorphic (Isomorphic quotient groups).

However, if the least size of a minimal generating set of each $$H_i$$ is the same as that of $$G$$, and $$H_1 \cong H_2$$, does this implies that $$G/H_1 \cong G/H_2$$?

• Do you really mean "same number of minimal generating sets", or do you mean "the least size of a minimal generating set of $H_i$ is the same as that of $G$"? Jul 24, 2023 at 14:10
• (The reason I ask is that $\mathbb{Z}_6$ has $1$ as the smallest size of a minimal generating set, but has four "minimal generating sets", namely $\{1\}$, $\{5\}$, $\{2,3\}$, and $\{3,4\}$). Jul 24, 2023 at 14:13
• yes, @ArturoMagidin, thank you! Jul 24, 2023 at 14:17
• @BrevanEllefsen unfortunately $\Bbb Z_3$ and $\Bbb Z_2$ are not isomorphic Jul 24, 2023 at 14:20

There are two subgroups isomorphic to $$C_2 \times C_4$$ in $$C_4 \times C_8$$ with non-isomorphic quotient groups.