1. Finite Groups and Subgroups, Terminology
At this time we are mainly concerned with finite groups, that is, groups with a finite number of elements.
The order of a group, |G|, is the number of elements in the group. The order of a group may be finite or infinite.
The order of an element, |a|, is the smallest positive integer n such that an = e.
The order of an element may likewise be finite or infinite.
Note: if |a|=2 then a=a-1. If |a|=1 then a=e.
A subgroup H of a group G is a subset of G together with the group operation, such that H is also a group.
That is, H is closed under the operation, and includes inverses and identity.
(Note: H must use the same group operation as G. So Zn, the integers mod n, is not a subgroup of Z, the integers, because the group operation is different.)
euler portrait

2. Cancellation and Conjugation
In any group,
a*b=a*c implies that b=c and
c*a=b*a implies that c=b.
This is used in proofs.
To conjugate an element a by x means to multiply thus:
xax-1 or x-1ax
While conjugating an element may change its value, the order
|a| is preserved.
This is useful in proofs and in solving matrix equations.
cancellation and conjugation

3. “Socks and Shoes” Property
When taking inverses of two or more elements composed together, the positions of the elements reverse.
That is, (a*b)-1 = b-1*a-1. For more elements, this generalizes to
(ab...yz)-1 = z-1y-1...b-1a-1.
In Abelian groups, it is also true that (ab)-1 = a-1b-1 and (ab)n = anbn. This also generalizes to more elements.
This is called the “socks and shoes property” as a mnemonic, because the inverse of putting on one's socks and shoes, in that order, is removing ones shoes and socks, in that order.
shoes and socks in the car
shoes and socks

4. The One Step Subgroup Test
Subgroup Tests:
The One Step Subgroup Test
Suppose G is a group and H is a non-empty subset of G.
If, whenever a and b are in H,
ab-1 is also in H,
then H is a subgroup of G.
Or, in additive notation:
a - b is also in H,
To apply this test:
Note that H is a non-empty subset of G.
Show that for any two elements
a and b in H, a*b-1 is also in H.
Conclude that H is a subgroup of G.
ab-1 H
Example: Show that the even integers are a subgroup of the Integers.
Note that the even integers is not an empty set because 2 is an even integer.
Let a and b be even integers.
Then a = 2j and b = 2k for some integers j and k.
a + (-b) = 2j + 2(-k) = 2(j-k) = an even integer
Thus a - b is an even integer
Thus the even integers are a subgroup of the integers.
one step at a time by norby

5. To Apply the Two Step Subgroup Test:
Subgroup Tests:
The Two Step Subgroup Test
Let G be a group and H a nonempty subset of G. If a●b is in H whenever a and b are in H, and a-1 is in H whenever a is in H, then H is a subgroup of G.
To Apply the Two Step Subgroup Test:
Note that H is nonempty
Show that H is closed with respect to the group operation
Show that H is closed with respect to inverses.
Conclude that H is a subgroup of G.
Example: show that 3Q*, the non-zero multiples of 3n where n is an integer, is a subgroup of Q*, the non-zero rational numbers.
3Q* is non-empty because 3 is an element of 3Q*.
For a, b in 3Q*, a=3i and b=3j where i, j are in Q*.
Then ab=3i3j=3(3ij), an element of 3Q* (closed)
For a in 3Q*, a=3j for j an element in Q*.
Then a-1=(j-1*3-1), an element of 3Q*. (inverses)
Therefore 3Q* is a subgroup of Q*.

6. The Finite Subgroup Test
Subgroup Tests:
The Finite Subgroup Test
Let H be a nonempty finite subset of G. If H is closed under the group operation, then H is a subgroup of G.
To Use the Finite Subgroup Test:
If we know that H is finite and non-empty, all we need to do is show that H is closed under the group operation. Then we may conclude that H is a subgroup of G.
Example: To Show that, in Dn, the rotations form a subgroup of Dn:
Note that the set of rotations is non-empty because R0 is a rotation.
Note that the composition of two rotations is always a rotation.
Therefore, the rotations in Dn are a subgroup of Dn.
math cartoons from

7. Examples of Subgroups: cyclic subgroups
Let G be a group, and a an element of G.
Let <a> = {an , where n is an integer},
(that is, all powers of a.)
...Or, in additive notation,,,
let <a>={na, where n is an integer},
(that is, all multiples of a.)
Then <a> is a subgroup of G.
Note:
In multiplicative notation, a0 = 1 is the identity; while 0a=0 is the identity in additive notation.
Thus <a> includes the identity.
Also note that the integers less than 0 are included here, so <a> includes all inverses.
For example:
In R*, <2>, the powers of 2, form a subgroup of R*.
In Z, <2>, the even numbers, form a subgroup.
In Z8, the integers mod 8, <2>={2,4,6,0}
is a subgroup of Z8.
In D3, the dihedral group of order 6, <R120> = {R0, R120, R240} is a subgroup of D3
Each element generates its own cyclic subgroup.
subgroup image

8. Examples of Subgroups: The Center of a Group Z(G)
The Center of a group, written Z(G), is the subset of elements in G which commute with all elements of G.
If G is Abelian, then Z(G)=G.
If G is non-Abelian, then Z(G) may consist only of the identity, or it may have other elements as well.
For example, Z(D4) = {R0, R180}.
The Center of a Group is a Subgroup of that group.
Subgroup lattice for D3

9. Examples of Subgroups: The Centralizer of an Element C(a)
For any element a in G, the Centralizer of a, written C(a)
is the set
of all elements of G
which commute with a.
In an Abelian group,
C(a) is the entire group. In a non-Abelian group, C(a) may consist only of the identity, a, and a-1,
or it may include other elements as well.
For example, in D3,
C(f) ={f, R0},
while C(R0)=D3
For each element a in a group G,
C(a) = the centralizer of a is a subgroup of G.
subgroup image