Tour:Subgroup

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This article adapts material from the main article: Subgroup

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Abelian group| UP: Introduction one (beginners)| NEXT: Trivial group
Expected time for this page: 10 minutes
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
PREREQUISITES: Definition of group. Return to group if you do not remember this.
WHAT YOU NEED TO DO:
  • Understand thoroughly, the definition in terms of closure of binary operation and the universal algebraic definition of subgroup
  • Read, and make sense, of the other definitions
  • Make sense of the examples given and use these to get some intuition about what being a subgroup means

WHAT YOU DO NOT NEED TO DO: Prove that all the definitions are equivalent. This will be covered in part two of the tour.

IF YOU ARE FOLLOWING ANOTHER PRIMARY TEXT: Compare the definition of subgroup given here, with the definition in your primary text

Definition

Definition in terms of closure under binary operation

This definition of subgroup corresponds to the textbook definition of group.

Let G be a group. A subset H of G is termed a subgroup if the following two conditions hold:

  • Whenever a,b belong to H, the product ab belongs to H.
  • With this induced multiplication, H becomes a group in its own right (i.e., it has an identity element, and every element has a two-sided inverse). Note that associativity in H follows automatically from associativity in G.

The universal algebraic definition

This definition of subgroup corresponds to the universal algebraic definition of group.

Let G be a group. A subset H of G is termed a subgroup if all the three conditions below are satisfied:

  • Whenever a, b belong to H, so does ab (here ab denotes the product of the two elements)
  • e belongs to H (where e denotes the identity element)
  • Whenever a belongs to H, so does a^{-1} (the multiplicative inverse of a)

Definition via the subgroup condition

The equivalence of this definition with the earlier one is often called the subgroup condition. For full proof, refer: Sufficiency of subgroup condition

It has two forms (left and right):


Notation

If H is a subgroup of G, we typically write H \le G or G \ge H. Some people also write H \subseteq G, but the latter notation is typically used for arbitrary subsets that need not be subgroups.

If H is not equal to the whole of G, we say that H is a proper subgroup of G, and this is sometimes denoted by H < G or G > H.

Examples

Examples in abelian groups

If we consider the abelian group (\R,+) (reals under addition) then the group of integers (\mathbb{Z},+) is a subgroup of this group. Similarly, the group of rational numbers (\mathbb{Q},+)) is an example of a subgroup of the group of reals.

On the other hand, the set of positive integers is not a subgroup of the group of integers, although it is closed under the group operation. This is because the additive inverse (or negative) of a positive integer isn't a positive integer.

PONDER (WILL BE EXPLORED LATER IN THE TOUR):
  • Over why the different definitions of subgroup are equivalent.
  • Over what parts of the definition of group are needed to prove the equivalence of definitions
WHAT'S MORE: Some more definitions, examples, and general information about subgroups. Some of it may use terminology that you haven't encounutered so far; ignore those parts.
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Abelian group| UP: Introduction one (beginners)| NEXT: Trivial group
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
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