# New outer automorphism for $G_1 \times G_2$

Suppose $$G_1$$ is a group, which has no outer automorphism.

Suppose $$G_2$$ is a group, which has no outer automorphism.

Main Question: What are the possible conditions to know can there be an outer automorphism for $$G_1 \times G_2?$$

An easier question:

Suppose $$G_1=SU(2)$$ is a group, which has no outer automorphism.

Suppose $$G_2= \mathbb{Z}/2\mathbb{Z}$$ is a group, which has no outer automorphism.

• $$SU(2) \times SU(2)$$ has however a order-2 outer automorphism.
• $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$$ has however a order-2 outer automorphism.

Simpler Question: Is there an outer automorphism for $$SU(2) \times \mathbb{Z}/2\mathbb{Z} ?$$

If you can answer anything of the above two Questions - this counts as a perfect answer.

A sufficient condition is that $$G_1$$ has a central subgroup $$C$$ isomorphic to a quotient of $$G_2$$ - this holds for your example.
We then have a natural map $$q:G_2\to C$$.
The map $$\phi:G_1\times G_2\to G_1\times G_2:(g,h)\mapsto (gq(h),h)$$ is an outer automorphism.
• just check the definition of automorphism. Injective: $\phi(g,h)=(1,1)$ then immediately $h=1$ so $q(1)=1$ gives $g=1$. Surjective: $(g,h)=\phi(gq(h)^{-1},h)$. Homomorphism: A little long for a comment, but straightforward – Robert Chamberlain Feb 25 at 16:04
• Should it be intead a natural map $q:G_2\to C$. The map $\phi:G_1\times G_2\to G_1\times G_2:(g,h)\mapsto (gq(h),h)$ is an outer automorphism??? – wonderich Feb 26 at 15:55