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**Introduction to Algebraic Groups**

One of the most fundamental algebraic structures in mathematics is the group.

A group is a set of elements paired with an operation that satisfies the following four conditions:

I. It is closed under an operation (represented here by “*+*”, although it does not necessarily mean addition):

For all elements *a* and *b* in the set S, *a+b* is also in S.

II. It contains an identity element (often written as “*e*”):

There is some element *e* in the set S such that for every element a in S,* a+e = e+a = a*.

III. The operation is associative:

For all *a*, *b*, and *c* in the set S, *(a+b)+c = a+(b+c)*.

IV. Inverses exist:

For every element *a* in the set S, there is an *a ^{-1}* in S such that

*a+a*We will look at some examples of groups and sets that aren’t groups.

^{-1}= a^{-1}+a = eExample 1. Integers under addition, (ℤ, +).

I. The group operation is addition. For any two integers *n* and *m*, *n+m* is also an integer.

II. The identity element is *0* because for any integer *n*, *n+0 = 0+n = n*.

III. Addition is associative.

IV. Inverses exist. The inverse of any integer *n* is *-n*, and *n + (-n) = (-n) + n = 0*. Therefore, (ℤ, +) is a group.

Example 2. Integers under multiplication, (ℤ, ⋅)

I. The group is closed under multiplication, because for any two integers *n* and *m*, *n⋅m* is also an integer.

II. The identity element is *1*. For any integer *n*,* n⋅1 = 1⋅n = n*.

III. Multiplication is associative.

IV. Inverses do NOT exist. For any integer *n*, *1/n* is not an integer except when *n=1*. Since the the set does not meet all four criteria, (ℤ, ⋅) is not a group.

Example 3. The set of rational numbers not including zero, under multiplication (ℚ – *0*, *⋅*)

I. The group is closed under multiplication.

II. *1* is the identity element.

III. Multiplication is associative.

IV. Inverses exist in this group.

If *q* is a rational number, *1/q* is also a rational number. And *q⋅ (1/q) = (1/q) ⋅ q = 1*.

Notice that if *0* were in this set, it would not be a group because *0* has no inverse. Therefore, ℚ – *0* is a group.

Example 4. The set {*0, 1, 2*} under addition

I. This set is not closed under addition because 1 + 2 = 3, and 3 is not part of the set. Therefore, the set cannot be a group.

Example 5. The trivial group {*e*} This group consists only of the identity element. We don’t need to specify the operation here because it works for both multiplication and addition.

For addition, *e=0. 0+0=0* so the group is closed under addition, has an identity element, and is closed under inverses (and addition is associative).

For multiplication, *e=1* and similarly,* 1⋅1 = 1*.

The examples here involved only numbers, but there are many different types of groups.

For example, the ways to transform a triangle is called a Dihedral Group, with the group operation being the act of rotating or reflecting the shape around an axis. The possible permutations of a Rubik’s Cube are also a group, with the operation being a sequence of moves. It’s powerful to know that a set is a group because it gives you an understanding of how the elements will always behave. Groups have many other properties that are useful to mathematicians, and there is a whole field of study built off of this knowledge called Group Theory.

The set of permutations of a Rubik’s Cube is considered an algebraic group.