## Review of Groups: Definitions and Basics

**Point of post: **In this post we will review the very basic definitions of group, subgroup, etc. This isn’t supposed to be a full-fledged discussion of group theory, since it’s assumed the reader will have already taken a course in algebra. This is just a quick refresher and something to get notations in sync. It covers just the basic definitions of groups and subgroups along with a few perfunctory theorems

*Groups*

A *group* is formally an ordered pair where is a set and is a mapping which satisfies the following axioms:

A group is called *abelian *if in addition to the above axioms it satisfies for all .

It is common to forgo the formalities of above and denote a group by just its underlying set and denote its group operation by concatenation or, if is abelian by addition. The element denoted in is known as the *identity element* and is denoted as if is abelian. The element given in is known as the *inverse of *and is denoted and if is abelian* . *A group containing one element (the identity element) is called *trivial*. The cardinality is called the *order *of* *and is denoted . If we call a *finite group. *For we may define arbitrary products of exponentially by and , and additively and . Similarly, we may define as and .

If and the set is non-empty we call the *order *of and denote this or when it’s necessary to distinguish between the order of a group and the order of an element. If then we say that is of *infinite order *and denote this by writing .

* * Some common theorems which I will state but not prove (they can be found in any book on algebra) are

**Theorem: ** *The identity element of a group is unique.*

**Theorem: ***The inverse of an element of a group is unique.*

**Theorem: ***The cancellation law holds, in the sense that if is a group and then implies .*

**Theorem (Generalized Associativity): ***Said simply, any arbitrary placement of parentheses does not change the value of a product of group elements. So, for example and so we may write an arbitrary product without parentheses and not be ambiguous.*

*Subgroups*

Let be a group and . Then, if is a group under the same binary operation as then we call a *subgroup *of and denote this by saying that . We call a *proper subgroup *if and and denote this . We call a *trivial subgroup *if it’s trivial. Thus, the non-trivial proper subgroups of are the such that .

We prove a few basic facts about subgroups

**Theorem: ***Let be a group and . Then, .*

**Proof: **Evidently since we have that there is some and so by assumption . We note then that if then . Lastly, we notice that the associativity of the multiplication is inherited from .

The converse is clear.

**Theorem: ***Let be a group and a class of subgroups. Then, *

**Proof: **We note that clearly since for every . We note next then that if then for every and thus for every and so from where the conclusion follows by the previous theorem.

**Corollary: ***Let be a group and . There exists a unique subgroup of which contains and is contained in any subgroup containing . In essence, there is a ‘smallest’ subgroup containing . *

**Proof: **Merely take .

is called the subgroup *generated *by .

From this we can get the following theorem which takes the almost intractable definition of the above idea of generated groups and shows how simple it really is.

**Theorem: ***Let be a group and , then *

**Proof: **This is clear since evidently the right hand side of the above is a subgroup of containing and any subgroup of containing must contain the right hand side of the above.

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

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