Section 6.4 Subgroups Section 6.4 Subgroups Section 6.4 Subgroups Section 6.4 Subgroups 629 Section 6.4 Subgroups : Groups Inside a Group Section 6.4 Subgroups : Groups Inside a Group Section 6.4 Subgroups : Groups Inside a Group Section 6.4 Subgroups : Groups Inside a Group Purpose of Section Purpose of Section Purpose of Section Purpose of Section To introduce the concept of a subgroup subgroup subgroup subgroup and find the subgroups of various symmetry groups. Introduction Introduction Introduction Introduction Recall the six symmetries of an equilateral triangle; the identity map, three flips about the midlines through the vertices of the triangle, and two (counterclockwise) rotations of 120 and 240 degrees. Symmetries of an equilateral triangle Figure 1 This set, along with the group operation of composition, forms a self- contained algebraic system called a group. It is distinguished by the fact the group operation is closed and the group contains an identity (do nothing operation), and every element in the group has an inverse. But this group is only the outside of the shell, inside there may be smaller groups. For example, in the dihedral group 3 D of six symmetries of an equilateral triangle, consider the subset of three rotational symmetries, the identity map e and the two rotations of 120 and 240 degrees. The Cayley table for these symmetries { } 120 240 , , eR R is drawn in Figure 2, which can easily be verified to form a group. The group operation is closed (i.e. the product of two elements belongs to the group), e is the identity, and each element has an inverse.
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Section 6.4 Subgroups : Groups Inside a GroupSection 6.4 Subgroups : Groups Inside a GroupSection 6.4 Subgroups : Groups Inside a GroupSection 6.4 Subgroups : Groups Inside a Group
Purpose of SectionPurpose of SectionPurpose of SectionPurpose of Section To introduce the concept of a subgroupsubgroupsubgroupsubgroup and find the
subgroups of various symmetry groups.
IntroductionIntroductionIntroductionIntroduction
Recall the six symmetries of an equilateral triangle; the identity map, three
flips about the midlines through the vertices of the triangle, and two
(counterclockwise) rotations of 120 and 240 degrees.
Symmetries of an equilateral triangle
Figure 1
This set, along with the group operation of composition, forms a self-
contained algebraic system called a group. It is distinguished by the fact the
group operation is closed and the group contains an identity (do nothing
operation), and every element in the group has an inverse. But this group is
only the outside of the shell, inside there may be smaller groups. For
example, in the dihedral group 3
D of six symmetries of an equilateral triangle,
consider the subset of three rotational symmetries, the identity map e and the
two rotations of 120 and 240 degrees. The Cayley table for these symmetries
{ }120 240, ,e R R is drawn in Figure 2, which can easily be verified to form a
group. The group operation is closed (i.e. the product of two elements
belongs to the group), e is the identity, and each element has an inverse.
Subgroup of rotations of symmetries of an equilateral triangle
Figure 2
This motivates the following definition of “groups within groups,” or
subgroups.
Definition:Definition:Definition:Definition: Let ( ),G ∗ be a group with operation ∗ . If a subset H G⊆ itself
forms a group with the same operation ∗ , then H is called a subgroupsubgroupsubgroupsubgroup of G .
If H is neither the identity { }e nor the entire group G , which are groups
called trivial subgroups trivial subgroups trivial subgroups trivial subgroups of G , then H is called a proper subgroupproper subgroupproper subgroupproper subgroup of .G
Example 1 (SubgroupExample 1 (SubgroupExample 1 (SubgroupExample 1 (Subgroups of Symmetries of an Equilateral Triangle)s of Symmetries of an Equilateral Triangle)s of Symmetries of an Equilateral Triangle)s of Symmetries of an Equilateral Triangle)
Find the proper subgroups of the dihedral group 3
D the symmetries of
an equilateral triangle.
Solution:Solution:Solution:Solution: The Cayley table for the dihedral group3
D of symmetries of an
equilateral triangle and its proper subgroups are displayed in Figure 3. There
are four proper subgroups of3
D ; the rotational subgroup { }120 240, ,e R R of order
3 and three “flip” subgroups { } { } { }, , , , ,v ne nw
Theorem Theorem Theorem Theorem 1111 (Conditions for Being a Subgroup) (Conditions for Being a Subgroup) (Conditions for Being a Subgroup) (Conditions for Being a Subgroup) Let ( ),G ∗ be a group with
operation ∗ and H a nonempty subset of G . The set H with operation ∗ is
a subgroupsubgroupsubgroupsubgroup ( ),H ∗ of ( ),G ∗ if the following two conditions hold:
i) H is closedclosedclosedclosed under ∗ . That is, ,x y H x y H∀ ∈ ⇒ ∗ ∈ .
ii) Every element in H has an inverseinverseinverseinverse in H . That is
( )( )( )1 1 1h H h H h h h h e
− − −∀ ∈ ∃ ∈ ∗ = ∗ = .
where " "e is the identity element in G .
Proof: Proof: Proof: Proof:
Since ∗ is a binary operation on G , it is also a binary operation on the
subset H , and by assumption i) we know ∗ maps H H× into H . Next, the
associative law ( ) ( )a b c a b c∗ ∗ = ∗ ∗ holds for all , ,a b c H∈ since H is a
subset of G and we know it holds for all , ,a b c G∈ . We now ask if the identity
e G∈ also belongs to H and is the identity of H ? The answer is yes since
by picking an h H∈ we know by hypothesis ii) there exists a 1h H−∈ , and by
closure 1h h e H−∗ = ∈ . Hence, we have verified the four properties required
for a group: closure, associativity, identity, and inverse. Hence H is a group.
▌
Example 2 (Test of Subgroup)Example 2 (Test of Subgroup)Example 2 (Test of Subgroup)Example 2 (Test of Subgroup) Let { }0, 1, 2,...G = = ± ±� be the group of
integers with the binary operation of addition + . Show the even integers
{ }2 0, 2, 4,...= ± ±� is a subgroup of G .
SolutionSolutionSolutionSolution
We observe that + is closed binary operation in 2� since if
1 22 , 2m k n k= = are even integers, so is their sum ( )1 2
2 2m n k k+ = + ∈ � .
Secondly, every even integer 2 2k ∈ � has an inverse, namely
( )2 2 2k k− = − ∈ � . ▌
Note:Note:Note:Note: The order of any subgroup of a group is a divisor of the group, and if the
order of the subgroup is a prime number then there will be a subgroup of that order.
Hence, there is not a subgroup of order 9 of a subgroup of order 30, and there
might be a subgroup of order 15, and there is a subgroup of order 5.
Example Example Example Example 3333 (Group of Infinite Order) (Group of Infinite Order) (Group of Infinite Order) (Group of Infinite Order) Let ( ),G ∗ be the group of points in the
plane 2� where the group operation :+ × →� � � is coordinate wise addition
of points ( ) ( ) ( ), , ,a b c d a c b d+ = + + . We leave it to the reader to show ( )2, +�
is a group. Show that the x -axis ( ){ },0 :H x x= ∈� is a subgroup of ( )2, +� .
SolutionSolutionSolutionSolution
The x -axis is a subset of the plane and the operation + is closed in H
since
( ) ( ) ( )1 2 1 2,0 , ,0 ,0x H x H x x H∈ ∈ ⇒ + ∈
Also every ( )1,0x H∈ has an inverse ( )1
,0x H− ∈ , i.e. ( ) ( ) ( )1 1,0 ,0 0,0x x+ − = ,
which is the group identity in 2� ▌
In general it is not a simple task to find all subgroups of a group, but for
cyclic groups it is an easy task.
Example Example Example Example 4444 (S (S (S (Subgroups of ubgroups of ubgroups of ubgroups of the Dihedral Groupthe Dihedral Groupthe Dihedral Groupthe Dihedral Group 4D )))) Figure 5 shows the dihedral
group 4
D of eight symmetries of a square, also called the octic octic octic octic group.
a) Is the octic group commutative? Hint: Compare products 270 ne
R F
and270
ne
F R .
b) There are several subsets of the eight symmetries that form a group
in their own right. These are called subgroups of the octic group. Can you
Note:Note:Note:Note: You may have noticed that the order of the subgroups seems to
always divide the order of the group. This is not a coincidence. The order of
a subgroup always divides the order of a group. For example a group of order
11 will only have the trivial subgroups of the group itself and the identity
subgroup. On the other hand the groups of order 6 we have seen (cyclic
group of order six and the dihedral group 3
D of symmetries of an equilateral
triangle both have subgroups of order 2 and 3.
Example Example Example Example 5555 (Subgroup Generated by (Subgroup Generated by (Subgroup Generated by (Subgroup Generated by 120
R ))))
Find the subgroup of the dihedral group 3
D of symmetries of an
equilateral triangle generated by 120
R .
SolutionSolutionSolutionSolution
Starting with 120
R and the identity 0
e R= we form the set { }0 120,R R after
which we compute 2
120 240R R= . Since this is not in { }0 120
,R R we include it,
getting{ }0 120 240, ,R R R . We now compute the next power 3
120 0R R= in which case
we stop, getting the subgroup { }120 0 120 240, ,R R R R= of rotations of
3D .
Example Example Example Example 6666 (Subgroups of a Cyclic Group) (Subgroups of a Cyclic Group) (Subgroups of a Cyclic Group) (Subgroups of a Cyclic Group) Find the subgroups of 8
Z
SolutionSolutionSolutionSolution
Systematically trying different generators, we find the 8 subgroups.
6. (Generated Groups of Symmetries of a Rectangle)(Generated Groups of Symmetries of a Rectangle)(Generated Groups of Symmetries of a Rectangle)(Generated Groups of Symmetries of a Rectangle) In the Klein 4-group
{ }180, , ,e R v h of symmetries of a rectangle, find the subgroups generated by
each element in the group. What is the order of each member?
Ans:Ans:Ans:Ans:
{ }
{ }
{ }
180 180 180 has order 2
has order 2
has order 2
,
,
,
R e R R
v e v v
h e h h
=
=
=
7. (Center of a Group)(Center of a Group)(Center of a Group)(Center of a Group) The center ( )Z G of a group G consists of all
elements of the group that commute with all elements of the group. That is
( ) { }: for all Z G g G gx xg x G= ∈ = ∈
It can be shown that the center of any group is a subgroup of the group. Find
the center of the group of symmetries of a rectangle. Note: The center of a
group is never empty since the identity element of a group always commutes
with every element of the group. The question is, are there other elements
that commute with every element of the group.
Ans:Ans:Ans:Ans: The center of the Klein 4-group is { }180,e R
8. (Hasse Diagram)(Hasse Diagram)(Hasse Diagram)(Hasse Diagram) Draw a Hasse diagram for the subgroups of symmetries