**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. Example 1. Integers under addition, (ℤ, +). I. The group operation is addition. For any two integers

^{-1}= a^{-1}+a = e*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.

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