4
$\begingroup$

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.

$\endgroup$
  • 4
    $\begingroup$ 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
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
7
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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
4
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.