TITLE: Why do we use G, H, and 'K' for groups?
QUESTION [8 upvotes]: In many cases of mathematics, we tend to set several variables in ascending alphabetical order.

$x,y,z \dots$ for normal variables
$A,B,C\dots$ for sets, coefficients, matrices
$V,W\dots$ for vector spaces
${I,J}\dots$ for indices, ideals
$\mathcal{C,D}\dots$ for categories

But when we need 3 groups, we tend to use $G, H$ and suddenly, $K$, Which is unexpected.
Here are some examples that groups are denoted by $K$.

Prove that quotient group K/H is normal subgroup of quotient group G/H
https://en.wikipedia.org/wiki/Group_homomorphism#The_category_of_groups
https://proofwiki.org/wiki/Pullback_of_Quotient_Group_Isomorphism_is_Subgroup

Why do we denote groups by $K$ regardless of the alphabetical order?

REPLY [5 votes]: I think that $K$ is also used because of the German word kompakt for a compact group. Consider for example the Iwasawa decomposition of a group $G$, which is
$$G=KAN,$$
where $G$ is a connected semisimple real Lie group, $K$ a compact subgroup, $A$ an abelian subgroup and $N$ a nilpotent group. So $A$ for an abelian group is also customary, as is $N$ for a nilpotent group.
On the other hand, $H$ as a group notation doesn't rely on a property, but rather that we need two groups in general for a group homomorphism
$$
\phi\colon G\rightarrow H,
$$
and mathematicians then just proceeded alphabetically.