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

Question

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.

Answer 1

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
• Rotman's Advanced Modern Algebra
• Herstein's Topics in Aglebra
• Dummit and Foote's 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.

Answer 2

I would say this $$H \subseteq K \subseteq G$$ and that $H$ is a subgroup of $K$ and $K$ is a subgroup of $G$.