Definition of Groups
Binary Operation |
Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G. |
Group |
Let G be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is a group under this operation if the following three properties are satisfied. |
Properties to Satisfy (Group) |
1. Closure 2. Associativity 3. Identity 4. Inverses |
Abelian group |
If a group has the property that ab = ba for every pair of elements a and b, we say the group is Abelian. |
Associativity |
The operation is associative; that is, (ab)c = a(bc) for all a, b, c in G. |
Identity |
There is an element e (called the identity) in G such that ae = ea = a for all a in G. |
Inverses |
For each element a in G, there is an element b in G (called an inverse of a) such that ab = ba = e. |
Modular Arithmetic |
When a = qn + r, where q is the quotient and r is the remainder upon dividing a by n, we write a mod n = r. |
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Elementary Properties of Groups
Theorem 2.1 Uniqueness of the Identity |
In a group G, there is only one identity element. |
Theorem 2.2 Cancellation |
In a group G, the right and left cancellation laws hold; that is, ba = ca implies b = c, and ab = ac implies b = c. |
Theorem 2.3 Uniqueness of Inverses |
For each element a in a group G, there is a unique element b in G such that ab = ba = e. |
Theorem 2.4 Socks-Shoes Property |
For group elements a and b, (ab)-1 = b-1a-1. |
Subgroup Tests
One-Step Subgroup Test |
Let G be a group and H a nonempty subset of G. If ab-1 is in H whenever a and b are in H, then H is a subgroup of G. (In additive notation, if a - b is in H whenever a and b are in H, then H is a subgroup of G.) |
Two-Step Subgroup Test |
Let G be a group and let H be a nonempty subset of G. If ab is in H whenever a and b are in H (H is closed under the operation), and a-1 is in H whenever a is in H (H is closed under taking inverses), then H is a subgroup of G. |
Theorem 3.3 Finite Subgroup Test |
Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G. |
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Examples of Subgroups
Theorem 3.4 <a> Is a Subgroup |
Let G be a group, and let a be any element of G. Then, <a> is a subgroup of G. |
Center of a Group |
The center, Z(G), of a group G is the subset of elements in G that commute with every element of G. In symbols,
Z(G) = {a ∈ G | ax = xa for all x in G}. |
Theorem 3.5 Center Is a Subgroup |
The center of a group G is a subgroup of G. |
Centralizer of a in G |
Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. In symbols,
C(a) = {g ∈ G | ga = ag}. |
Theorem 3.6 C(a) Is a Subgroup |
For each a in a group G, the centralizer of a is a subgroup of G. |
Terminology and Notation
Order of a Group |
The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G. |
Order of an Element |
The order of an element g in a group G is the smallest positive integer n such that gn = e. (In additive notation, this would be ng = 0.) If no such integer exists, we say that g has infinite order. The order of an element g is denoted by |g|. |
Subgroup |
If a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G. |
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