Question #23890

Let G ⊆ GLn(k) be a linear group over a field k. Show that every finite group can be realized as a linear group, but not every infinite group can be realized as a linear group.

Expert's answer

If *G *is a finite group, *G *actson itself faithfully by left multiplication, so we can embed *G *as agroup of permutation matrices in GL*m*(Z) *⊆** *GL*m*(Q), where *m *= *|G|*.

We finish by constructing aninfinite group*G *which cannot be realized as a linear group. Take anynonabelian finite group *H*. Since *H *is f.c., the direct sum *G *:= *H **⊕** H **⊕**· · · *is also f.c. However, since *H *not equal *Z*(*H*),

*Z*(*G*) = *Z*(*H*) *⊕** Z*(*H*)*⊕**· · · *has infinite index in *G*. By the firstpart of the exercise, *G *cannot be realized as a linear group.

We finish by constructing aninfinite group

## Comments

## Leave a comment