Is it necessary that $f: G_1 \to G_2$ is an isomorphism, for $f:H\to H$ an automorphism with $G_1, G_2\le H$ of the same cardinality?

Question

This might be a very trivial question, and I have explained what I think about it below. Let's say I have an automorphism $f : H \to H$. Now, let's say I take two subgroups $G_1$ and $G_2$ of $H$. Is it necessary that $f: G_1 \to G_2$ is also an isomorphism?

I think yes, cause all the elements of $G_1$ and $G_2$ are still the elements of H, and if $f$ is an automorphism, $f: G_1 \to G_2$ should necessarily be an isomorphism too.

Edit: Originally, I forgot to say that the two subgroups $G_1$ and $G_2$ have same cardinality.

Answer 1

If you're assuming that $\text{Im}(f|_{G_1}) \subseteq G_2$ (otherwise the question doesn't really make any sense), then the answer is yes if $G_1$ is finite and not necessarily otherwise.

If $G_1$ is finite, then since $f$ is monomorphism, $f : G_1 \to G_2$ is also a monomorphism and a monomorphism between two finite groups of the same cardinality is an isomorphism.

If $G_1$ can be infinite, we can take $H = \bigoplus_{n \in \mathbb{N}} \mathbb{Z}, \ G_1 = \{0\} \ \oplus \ \bigoplus_{n \geq 1}\mathbb{Z}, \ G_2=H$ and $f = 1_H$. Then $f : G_1 \to G_2$ is not surjective but $G_1$ and $G_2$ have the same cardinality.

Answer 2

No.

Let $G_1=\Bbb Z_6$, $G_2=D_{6}$, and $H$ be some suitably large $S_n$, $f=1_H$.

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NB: Here $D_6$ is the dihedral group of six elements.