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I was reading this page: http://www.proofwiki.org/wiki/Definition:Subgroup

I never heard that $H ≤G$ means $H$ is a subgroup of $G$. Is this standard notation ?

And if not, what is/are normal symbolic notations to say that $H$ is subgroup of $G$ ?

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Yes, that's standard. The collection of subgroups $H$ of a group $G$ form a lattice with partial order $\le$ given by inclusion. Some use $\subseteq$ besides $\le$, but I like to use the former for arbitrary subsets and so reserve the latter for subgroups. –  anon Mar 31 '13 at 19:51
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up vote 4 down vote accepted

Yes, this is pretty much "the standard": $$H\leq G \quad\iff \quad\text{ H is a subgroup of G}$$

In a sense, it is analogous to $H \subseteq G$, which is used to denote $H$ is a subset of $G$. $H\subseteq G$ is true for of any subgroup H of group G. If $H \leq G$, then it follows that $H\subseteq G$, but the converse does not hold, since groups have added algebraic structure. And so using the symbol $\subseteq$ to denote the relationship between $H$ and $G$ doesn't convey that in addition, $H$ is a subgroup of $G$ under the operation of $G$. $\quad H \leq G$ thus conveys MORE information than does $H \subseteq G$.

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I don't think you can find any other standard answer besides to the other answers.

Just two points:

  • If the subgroup $H$ of group $G$ is a proper one it is denoted by $H<G$.

  • In some books like books of J.J.Rotman in Group theory, you may see that he is using the standard symbol in another way. Especially when he is working on Solvable Groups. I mean the solvable series, e.g.: $$G=G_0\geq G_1\geq...\geq G_n=1$$

But, the standard is what anon noted.

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How is it being used in a different way in that example? –  Tobias Kildetoft Mar 31 '13 at 20:19
    
@TobiasKildetoft: I just noted that, I saw the way he used $\leq$. Different means to me is $\leq\to\geq$ and nothing is changed. –  B. S. Mar 31 '13 at 20:22
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