# Suppose $H,K$ are subgroups of $G$, what does it mean to write $H \leq K \leq G$?

I am following J.B. Fraleigh: A first course in abstract algebra. In the text page $101$ the author supposes $H,K$ are subgroups of $G$ then uses the notation $H \leq K \leq G$. Does he intend to write, $H \subseteq K \subseteq G$, $|H| \leq |K| \leq |G|$, or something different? I cannot recall that such notation for sets has been defined in my prior maths courses.

• I think it's subgroup notation. – user250285 Feb 14 '18 at 16:35
• What's that? I cannot recall the definition of subgroup notation. – Mikkel Rev Feb 14 '18 at 16:36
• This is pretty nonstandard notation, but the assumptions about what it means by others in here seem pretty reasonable. I suspect that this is not used much because it is a bit close to normal subgroup notation. – Cameron Williams Feb 14 '18 at 16:39
• For those of us who grew up on Herstein's Algebra, it is THE standard notation. – Lee Mosher Feb 14 '18 at 16:49
• I disagree with @Cameron, in my experience it is very common, even standard, to write $H\le G$ to say $H$ is a subgroup of $G$. – anon Feb 16 '18 at 1:25

As the other answer makes clear people will prefer different things. But it is completely standard to write $$H\leq G$$ when we want to say that $H$ is a subgroup of $G$. So saying that $$H\leq K\leq G$$ says that $H$ is a subgroup of $K$ and $K$ is a subgroup of $G$. (As a side note, $\leq$ is a transitive, so this would also mean that $H$ is a subgroup of $G$.)

I looked in a couple of my abstract algebra books, and the following books use $\leq$ for subgroups

• Gallian's Contemporary Abstract Algebra book
• Fraleigh's A First Course in Abstract Algebra
Even Wikipedia's article on subgroups uses the notation. Hungerford uses $<$.
One advantage of using $\leq$ over $\subseteq$ is that it distinguishes between being a subset and a subgroup. In a proof you might first show that $H$ is a subset of $G$ and then later conclude that $H$ is a subgroup. So having different notations can be helpful.
• It is also worth noting that $\leq$ is a natural notation since it is a partial ordering on the subgroup lattice of the group. – David Hill Feb 14 '18 at 18:28
I would say this $$H \subseteq K \subseteq G$$ and that $H$ is a subgroup of $K$ and $K$ is a subgroup of $G$.
• True, so I write that $H$ is also a subgroup of $K$ ... – Aqua Feb 14 '18 at 17:12