I have been thinking the relation between two stable groups formed as follow:

1)Gl(2,\R)\subseteq\GL(3,\R)\subseteq\cdots\GL(n,\R) G(\infty,\R)=\cup_{n=2}^{\infty}GL(n,\R) 2)GL(3,\R)\subseteq\GL(5,\R)\subseteq\cdots\GL(2n+1,\R) H(\infty,\R)=\cup_{n=1}^{\infty}GL(2n+1,\R)

It is clear that H(\infty,\R) is a subgroup of G(\infty,\R). Do you think that they might be isomorphic as a groups? If they are not, then what is the proper approach to explain them?

I can write a map from G(\infty,\R) to H(\infty,\R), for instance if $A\in GL(2,\R)$ and I want to send it to $GL(5,\R), then the map take A and send it to diag(A,I_{3}). This map injective homomorphism, but it is nor surjective!

I do not know maybe they are not isomorphic, but in this case how can convience myself?

Many thanks.

I have been thinking the relation between two stable groups formed as follow:

1. $GL(2,\mathbb R)\subseteq GL(3,\mathbb R)\subseteq\cdots GL(n,\mathbb R)$, $G(\infty,\mathbb R)=\cup_{n=2}^{\infty} GL(n,\mathbb R)$

2)$GL(3,\mathbb R)\subseteq GL(5,\mathbb R)\subseteq\cdots GL(2n+1,\mathbb R)$, $H(\infty,\mathbb R)=\cup_{n=1}^{\infty}GL(2n+1,\mathbb R)$

It is clear that $H(\infty,\mathbb R)$ is a subgroup of $G(\infty,\mathbb R)$. Do you think that they might be isomorphic as a groups? If they are not, then what is the proper approach to explain them?

I can write a map from $G(\infty,\mathbb R)$ to $H(\infty,\mathbb R)$, for instance if $A\in GL(2,\mathbb R)$ and I want to send it to $GL(5,\mathbb R)$, then the map take $A$ and send it to $diag(A,I_{3})$. This map injective homomorphism, but it is nor surjective!

I do not know maybe they are not isomorphic, but in this case how can convince myself?