Tour:Subgroup
This article adapts material from the main article: Subgroup
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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 be a group. A subset
of
is termed a subgroup if the following two conditions hold:
- Whenever
belong to
, the product
belongs to
.
- With this induced multiplication,
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
follows automatically from associativity in
.
The universal algebraic definition
This definition of subgroup corresponds to the universal algebraic definition of group.
Let be a group. A subset
of
is termed a subgroup if all the three conditions below are satisfied:
- Whenever
belong to
, so does
(here
denotes the product of the two elements)
-
belongs to
(where
denotes the identity element)
- Whenever
belongs to
, so does
(the multiplicative inverse of
)
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):
- A subset of a group is termed a subgroup if it is nonempty and is closed under the left quotient of elements. In other words, a subset
of a group
is termed a subgroup if and only if
is nonempty and
whenever
- A subset of a group is termed a subgroup if it is nonempty and is closed under the right quotient of elements. In other words, a subset
of a group
is termed a subgroup if and only if
is nonempty and
whenever
Notation
If is a subgroup of
, we typically write
or
. Some people also write
, but the latter notation is typically used for arbitrary subsets that need not be subgroups.
If is not equal to the whole of
, we say that
is a proper subgroup of
, and this is sometimes denoted by
or
.
Examples
Examples in abelian groups
If we consider the abelian group (reals under addition) then the group of integers
is a subgroup of this group. Similarly, the group of rational numbers (
) 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):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.
- 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
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