A group is a set of elements with a binary operation that satisfies four properties: closure, associativity, identity, and inverses. Groups can be finite or infinite, commutative or non-commutative. Examples of groups include integers under addition, rational numbers without 0 under multiplication, general linear groups of matrices, integers modulo n under addition, and dihedral and permutation groups. The defining properties of a group ensure the binary operation behaves similarly to ordinary arithmetic.

2. DEFINITION OF A GROUP
• A Group G is a collection of elements together with a binary
operation* which satisfies the following properties:
• Closure
• Associativity
• Identity
• Inverses
• .
* A binary operation is a function on G
which assigns an element of G to each
ordered pair of elements in G. For
example, multiplication and addition are
binary operations

3. CLASSIFICATION OF GROUPS
Finite or Infinite
Commutative or Non-Commutative,
(Commutative groups are also called
Abelian groups.)

4. EXAMPLES OF GROUPS
Infinite, Abelian:
The Integers under Addition (Z. +)
The Rational Numbers without 0 under multiplication (Q*,
X)
Infinite, Non-Abelian:
The General Linear Groups (GL,n), the nonsingular nxn
matrices under matrix multiplication

5. EXAMPLES OF GROUPS
Finite, Abelian:
The Integers Mod n under Modular Addition (Zn , +)
The “U groups”, U(n), defined as Integers less than n
and relatively prime to n, under modular multiplication.

6. EXAMPLES OF GROUPS
Finite, Non-Abelian:
The Dihedral Groups Dn the permutations on a regular
n-sided figure under function composition.
The Permutation Groups Sn, the one to one and onto
functions from a set to itself under function composition.

7. PROPERTIES OF A GROUP: CLOSURE
Example:
The Integers under Addition, (Z, +)
1 and 2 are elements of Z,
1+2 = 3, also an element of Z
Non-Examples:
The Odd Integers are not closed under Addition. For example, 3 and 5 are odd integers, but
3+5 = 8 and 8 is not an odd integer.
The Integers lack inverses under Multiplication, as do the Rational numbers (because of 0.)
However, if we remove 0 from the Rational numbers, we obtain an infinite closed group under
multiplication.

8. PROPERTIES OF A GROUP: ASSOCIATIVITY
The Associative Property, familiar from ordinary arithmetic on real numbers,
states that (ab)c = a(bc). This may be extended to as many elements as
necessary.
For example:
In Integers, a+(b+c) = (a+b)+c.
In Matrix Multiplication, (A*B)*C=A*(B*C).
In function composition,
f*(g*h) = (f*g)*h.
This is a property of all groups.

9. PROPERTIES OF A GROUP: IDENTITY
The Identity Property, familiar from ordinary arithmetic on real
numbers, states that, for all elements a in G, a+e = e+a = a.
For example, in Integers, a+0 = 0+a = a.
In (Q*, X), a*1 = 1*a = a.
In Matrix Multiplication, A*I = I*A = A.
This is a property of all groups.

10. PROPERTIES OF A GROUP: INVERSES
The inverse of an element, combined with that element, gives the identity.
Inverses are unique. That is, each element has exactly one inverse, and no two distinct elements have the same inverse.
The uniqueness of inverses is used in proofs.
For example...
In (Z,+), the inverse of x is -x.
In (Q*, X), the inverse of x is 1/x.
In (Zn, +), the inverse of x is n-x.
In abstract algebra, the inverse of an element a is usually written a-1.
This is why (GL,n) and (SL, n) do not include singular matrices; only nonsingular matrices have inverses.
In Zn, the modular integers, the group operation is understood to be addition, because if n is not prime, multiplicative
inverses do not exist, or are not unique.
The U(n) groups are finite groups under modular multiplication.