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.

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    $\begingroup$ I think it's subgroup notation. $\endgroup$ – user250285 Feb 14 '18 at 16:35
  • $\begingroup$ What's that? I cannot recall the definition of subgroup notation. $\endgroup$ – Mikkel Rev Feb 14 '18 at 16:36
  • $\begingroup$ 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. $\endgroup$ – Cameron Williams Feb 14 '18 at 16:39
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    $\begingroup$ For those of us who grew up on Herstein's Algebra, it is THE standard notation. $\endgroup$ – Lee Mosher Feb 14 '18 at 16:49
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    $\begingroup$ 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 $G$. $\endgroup$ – 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
  • 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.

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    $\begingroup$ It is also worth noting that $\leq$ is a natural notation since it is a partial ordering on the subgroup lattice of the group. $\endgroup$ – 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 $G$.

  • $\begingroup$ It takes more than being a subset of a set to make it a subgroup of a group. $\endgroup$ – Namaste Feb 14 '18 at 17:10
  • $\begingroup$ True, so I write that $H$ is also a subgroup of $K$ ... $\endgroup$ – Aqua Feb 14 '18 at 17:12

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