Let $A'$ be subgroup of $A$ and $B'$ be subgroup of $B$. What can we say if $A'/B' \cong A/B$

Question

Say $A'$ is subgroup of $A$ and $B'$ is subgroup of $B$ such that the quotient group $\frac{A'}{B'}$ is isomorphic to $\frac{A}{B}$. Does this imply any additional relationship between $A$ and $A'$ or $B$ and $B'$?

I can't seem to think of any, so help would be appreciated, thanks!

The answer here writes out explicitly a characterization of subgroups of a direct product using Goursat's lemma. It involves the sort of data you have in your situation. — Brian Shin, Jul 26 '21 at 15:43
I'm not really understanding the close votes here. The question seems to be asked out of interest - it certainly is unlikely to be an exam question, and likely to be of interest to others in the future. — user1729, Jul 27 '21 at 16:41

Dietrich Burde · Answer 1 · 2021-07-26T15:36:57.970

3

Already a trivial case like $A=B$ and $A'=B'$ being subgroups gives $A/B=A'/B'=1$, but we don't have any relationship between $A$ and $A'$, except for the assumption that $A'$ is a subgroup of $A$.

For examples with nontrivial quotient group consider $A=\Bbb Z$ and $B=2\Bbb Z$, with $A'=2\Bbb Z$ and $B'=4\Bbb Z$. Then $$ A'/B'=2\Bbb Z/4\Bbb Z\cong \Bbb Z/2\Bbb Z=A/B. $$

edited Jul 26 '21 at 15:36

answered Jul 26 '21 at 15:31

Dietrich Burde

130,978

Please refresh you understanding about the Enforcement of Quality Standards. First you answer a dupe, and then answer a PSQ, and ..... in one day! – amWhy Jul 27 '21 at 16:42
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@amWhy Yes, I understand (and have deleted my answer at the dupe). This question, however, deserves an answer, I believe. – Dietrich Burde Jul 28 '21 at 08:26

Let $A'$ be subgroup of $A$ and $B'$ be subgroup of $B$. What can we say if $A'/B' \cong A/B$

1 Answers1