# Analogies between finite groups and Lie groups

I believe there are some striking analogous facts between finite groups and Lie groups.

One analogue almost too basic to mention is the appropriate notion of subobjects. In elementary group theory the correct notion of subobject is just a subset closed under the operations (identity, inverses and composition). A Lie group is also a topological space, and the corresponding notion of closure is being closed under the operation of taking limits - so closed in the usual sense of topology, thus the corresponding notion of Lie subgroup is closed subgroup.

Here are some other notable analogies I can think of:

• A standard exercise in group theory asks us to prove that if $Z$ is central in $G$ then $G/Z$ cannot be cyclic (i.e. $\Bbb Z$ or $\Bbb Z/n\Bbb Z$). The analogue for Lie groups is that if a closed subgroup $Z$ is central then $G/Z$ cannot be of dimension $1$, so $\Bbb R$ or $\Bbb R/\Bbb Z$. Proof. The cyclic subgroups $C$ of $G/Z$ pull back to abelian subgroups of $G$, which commute elementwise pairwise and whose union is dense, making $G$ abelian.
• Fundamental Theorem of (appropriately "finite") AbGrps. For f.g. abelian groups, there is a surjection $\Bbb Z^n\to G$ for some $n$ making $G$ a quotient of $\Bbb Z^n$, which must be a direct product of quotients of $\Bbb Z$ (which are $\Bbb Z$ or $\Bbb Z/n\Bbb Z$). Similarly, for connected abelian Lie groups the exponential map $\exp:{\frak g}\to G$ should be a surjective group homomorphism so $G$ is a quotient of $\Bbb R^n$ (for some $n$) which must be a direct product of quotients of $\Bbb R$ (so $\Bbb R$ or $\Bbb R/\Bbb Z$).
• Orbit-Stabilizer: In finite group theory, if $G$ acts transitively on a set $X$ and there is a point-stabilizer $K$, then $|G|=|X|\cdot|K|$. In Lie theory, if $X$ is a homogeneous space with respect to a smooth action of $G$ and there is a point-stabilizer $K$, then $\dim G=\dim X+\dim K$. These are numerical versions of the OS Thm, a more structural flavor can be given with:
• Lagrange's Theorem. In elementary group theory, this essentially says $G/H$ is a partition of $G$, which in Lie theory we can express as $H\to G\to G/H$ being a fiber bundle.

Any other good ones we can add to this list, big or small?

• A much stronger analogy is between the classification of finite simple groups and of simple Lie groups; it's quite remarkable how close it is to being true that finite simple groups are all "of Lie type." – Qiaochu Yuan Nov 10 '15 at 21:51