Why do we use G, H, and 'K' for groups? 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?
 A: 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.
A: These are choices that mathematical writers have to make. There's not enough letters for all the stuff we need them to represent, and they often clash. So nice clean alphabetic sequences reserved for special purposes sometimes get in the way of our notational needs.
The letters $I$ and $J$ are also very commonly used for intervals in the real line. In particular $I$ is often reserved for one very special interval, namely $I = [0,1] \subset \mathbb R$. It is even somewhat common to be writing about groups and intervals at the same time, for example when writing about the fundamental group $G = \pi_1(X)$ of a topological space $X$ in which the elements of the group are represented by certain continuous paths $f : I \to X$ where $I = [0,1]$.
So, due to its many other busy duties, to avoid clash of variables the letter $I$ is rarely used for groups.
And $J$ usually refuses to be used too, in protest of the insult to his buddy $I$.
On the other hand $K$ is happy to be used for a group, particularly as a normal subgroup of a group $G$, because in that case $K$ is the Kernel of the quotient homomorphism $G \mapsto G / K$. And we sometimes jump wayyyyy further in the alphabet just to use $Q = G / K$ for the Quotient group.
