# Elementary group theory

A group must obey the following rules (or axioms). Let ${\displaystyle a,b,c}$ be arbitrary elements of ${\displaystyle G}$. Then:

An abelian group also obeys the additional rule:

## Notation

The group ${\displaystyle \{G,\perp \}}$ is often referred to as "the group ${\displaystyle G}$" or more simply as "${\displaystyle G.}$" Nevertheless, the operation "${\displaystyle \perp }$" is fundamental to the description of the group. ${\displaystyle \{G,\perp \}}$ is usually read as "the group ${\displaystyle G}$ under ${\displaystyle \perp }$". When we wish to assert that ${\displaystyle G}$ is a group (for example, when stating a theorem), we say that "${\displaystyle G}$ is a group under ${\displaystyle \perp }$".

The group operation ${\displaystyle \perp }$ can be interpreted in a great many ways. The generic notation for the group operation, identity element, and inverse of ${\displaystyle a}$ are ${\displaystyle \perp ,e,a',}$ respectively. Because the group operation associates, parentheses have only one necessary use in group theory: to set the scope of the inverse operation.

Group theory may also be notated:

Other notations are of course possible.

## Alternative Axioms

The pair of axioms A3 and A4 may be replaced either by the pair:

or by the pair:

These evidently weaker axiom pairs are trivial consequences of A3 and A4. We will now show that the nontrivial converse is also true. Given a left neutral element ${\displaystyle e,}$ and for any given ${\displaystyle a\in {}G,}$ then A4’ says there exists an ${\displaystyle x}$ such that ${\displaystyle x\perp {}a=e}$.

{\displaystyle {\begin{aligned}e&=y\perp (a\perp x)&\quad (1)\\&=y\perp (a\perp (e\perp x))&\quad (A3')\\&=y\perp (a\perp ((x\perp a)\perp x))&\quad (A4')\\&=y\perp (a\perp (x\perp (a\perp x)))&\quad (A2)\\&=y\perp ((a\perp x)\perp (a\perp x))&\quad (A2)\\&=(y\perp (a\perp x))\perp (a\perp x)&\quad (A2)\\&=e\perp (a\perp x)&\quad (1)\\&=a\perp x&\quad (A3')\\\end{aligned}}}

This establishes A4 (and hence A4”).

Proof.

{\displaystyle {\begin{aligned}a\perp e&=a\perp (x\perp a)&\quad (A4')\\&=(a\perp x)\perp a&\quad (A2)\\&=e\perp a&\quad (A4)\\&=a&\quad (A3')\\\end{aligned}}}

This establishes A3 (and hence A3”).

Theorem: Given A1 and A2, A3’ and A4’ imply A3 and A4.

Proof. Theorems 1.2 and 1.2a.

Theorem: Given A1 and A2, A3” and A4” imply A3 and A4.

Proof. Similar to the above.

## Basic theorems

### Identity is unique

Theorem 1.4: The identity element of a group ${\displaystyle \{G,\perp \}}$ is unique.

Proof: Suppose that ${\displaystyle e}$ and ${\displaystyle f}$ are two identity elements of ${\displaystyle G}$. Then

{\displaystyle {\begin{aligned}e&=&e\perp f&\quad (A3'')\\&=&f&\quad (A3')\\\end{aligned}}}

As a result, we can speak of the identity element of ${\displaystyle \{G,\perp \}}$ rather than an identity element. Where different groups are being discussed and compared, ${\displaystyle e_{G}}$ denotes the identity of the specific group ${\displaystyle \{G,\perp \}}$.

### Inverses are unique

Theorem 1.5: The inverse of each element in ${\displaystyle \{G,\perp \}}$ is unique.

Proof: Suppose that ${\displaystyle h}$ and ${\displaystyle k}$ are two inverses of an element ${\displaystyle g}$ of ${\displaystyle G}$. Then

{\displaystyle {\begin{aligned}h&=&h\perp e&\quad (A3)\\&=&h\perp (g\perp k)&\quad (A4)\\&=&(h\perp g)\perp k&\quad (A2)\\&=&e\perp k&\quad (A4)\\&=&k&\quad (A3)\\\end{aligned}}}

As a result, we can speak of the inverse of an element ${\displaystyle a}$, rather than an inverse. Without ambiguity, for all ${\displaystyle a}$ in ${\displaystyle G}$, we denote by ${\displaystyle a'}$ the unique inverse of ${\displaystyle a}$.

### Inverting twice takes you back to where you started

Equivalently, inverting is an involution.

### Inverse of ab

Proof. ${\displaystyle (a\perp b)\perp (b'\perp a')=a\perp (b\perp b')\perp a'=a\perp e\perp a'=a\perp a'=e}$. The conclusion follows from Theorem 1.4.

### Cancellation

Proof.
(1) If ${\displaystyle x=y}$, then multiplying by the same value on either side preserves equality.
(2) If ${\displaystyle a\perp x=a\perp y}$ then by (1)

{\displaystyle {\begin{aligned}&&a'\perp (a\perp x)&=&a'\perp (a\perp y)\\&\Rightarrow &(a'\perp a)\perp x&=&(a'\perp a)\perp y\\&\Rightarrow &e\perp x&=&e\perp y\\&\Rightarrow &x&=&y\\\end{aligned}}}

(3) If ${\displaystyle x\perp a=y\perp a}$ we use the same method as in (2).

### Latin square property

Theorem 1.3: For all elements ${\displaystyle a,b}$ in a group ${\displaystyle \{G,\perp \}}$, there exists a unique ${\displaystyle x\in G}$ such that ${\displaystyle a\perp x=b}$, namely ${\displaystyle x=a'\perp b}$.

### Powers

${\displaystyle a^{n}:={\begin{cases}\underbrace {a\perp {}a\perp \cdots \perp {}a} _{n\ {\text{times}}},&{\mbox{if }}n>0\\e,&{\mbox{if }}n=0\\\underbrace {a'\perp {}a'\perp \cdots \perp {}a'} _{-n\ {\text{times}}},&{\mbox{if }}n<0\end{cases}}}$
${\displaystyle {\begin{matrix}a^{m}\perp {}a^{n}&=&a^{m+n}\\(a^{m})^{n}&=&a^{m*n}\end{matrix}}}$

## Order

### Of a group element

The order of an element a in a group G is the least positive integer n such that an = e. Sometimes this is written "o(a)=n". n can be infinite.

Theorem 1.10: A group whose nontrivial elements all have order 2 is abelian. In other words, if all elements g in a group G g*g=e is the case, then for all elements a,b in G, a*b=b*a.

Proof. Let a, b be any 2 elements in the group G. By A1, a*b is also a member of G. Using the given condition, we know that (a*b)*(a*b)=e. Hence:

• b*a
• =e*(b*a)*e
• = (a*a)*(b*a)*(b*b)
• =a*(a*b)*(a*b)*b
• =a*e*b
• =a*b.

Since the group operation * commutes, the group is abelian.

### Of a group

The order of the group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G, in which case <G,*> is a finite group. If G is an infinite set, then the group <G,*> has order equal to the cardinality of G, and is an infinite group.

## Subgroups

A subset H of G is called a subgroup of a group <G,*> if H satisfies the axioms of a group, using the same operator "*", and restricted to the subset H. Thus if H is a subgroup of <G,*>, then <H,*> is also a group, and obeys the above theorems, restricted to H. The order of subgroup H is the number of elements in H.

A proper subgroup of a group G is a subgroup which is not identical to G. A non-trivial subgroup of G is (usually) any proper subgroup of G which contains an element other than e.

Theorem 2.1: If H is a subgroup of <G,*>, then the identity eH in H is identical to the identity e in (G,*).

Proof. If h is in H, then h*eH = h; since h must also be in G, h*e = h; so by theorem 1.8, eH = e.

Theorem 2.2: If H is a subgroup of G, and h is an element of H, then the inverse of h in H is identical to the inverse of h in G.

Proof. Let h and k be elements of H, such that h*k = e; since h must also be in G, h*h -1 = e; so by theorem 1.5, k = h -1.

Given a subset S of G, we often want to determine whether or not S is also a subgroup of G. A handy theorem valid for both infinite and finite groups is:

Theorem 2.3: If S is a non-empty subset of G, then S is a subgroup of G if and only if for all a,b in S, a*b -1 is in S.

Proof. If for all a, b in S, a*b -1 is in S, then

• e is in S, since a*a -1 = e is in S.
• for all a in S, e*a -1 = a -1 is in S
• for all a, b in S, a*b = a*(b -1) -1 is in S

Thus, the axioms of closure, identity, and inverses are satisfied, and associativity is inherited; so S is subgroup.

Conversely, if S is a subgroup of G, then it obeys the axioms of a group.

• As noted above, the identity in S is identical to the identity e in G.
• By A4, for all b in S, b -1 is in S
• By A1, a*b -1 is in S.

The intersection of two or more subgroups is again a subgroup.

Theorem 2.4: The intersection of any non-empty set of subgroups of a group G is a subgroup.

Proof. Let {Hi} be a set of subgroups of G, and let K = ∩{Hi}. e is a member of every Hi by theorem 2.1; so K is not empty. If h and k are elements of K, then for all i,

• h and k are in Hi.
• By the previous theorem, h*k -1 is in Hi
• Therefore, h*k -1 is in ∩{Hi}.

Therefore for all h, k in K, h*k -1 is in K. Then by the previous theorem, K=∩{Hi} is a subgroup of G; and in fact K is a subgroup of each Hi.

Given a group <G,*>, define x*x as x², x*x*x*...*x (n times) as xn, and define x0 = e. Similarly, let x -n for (x -1)n. Then we have:

Theorem 2.5: Let a be an element of a group (G,*). Then the set {an: n is an integer} is a subgroup of G.

A subgroup of this type is called a cyclic subgroup; the subgroup of the powers of a is often written as <a>, and we say that a generates <a>.

## Cosets

If S and T are subsets of G, and a is an element of G, we write "a*S" to refer to the subset of G made up of all elements of the form a*s, where s is an element of S; similarly, we write "S*a" to indicate the set of elements of the form s*a. We write S*T for the subset of G made up of elements of the form s*t, where s is an element of S and t is an element of T.

If H is a subgroup of G, then a left coset of H is a set of the form a*H, for some a in G. A right coset is a subset of the form H*a.

If H is a subgroup of G, the following useful theorems, stated without proof, hold for all cosets:

• Any x and y are elements of G, then either x*H = y*H, or x*H and y*H have empty intersection.
• Every left (right) coset of H in G contains the same number of elements.
• G is the disjoint union of the left (right) cosets of H.
• Then the number of distinct left cosets of H equals the number of distinct right cosets of H.

Define the index of a subgroup H of a group G (written "[G:H]") to be the number of distinct left cosets of H in G.

From these theorems, we can deduce the important Lagrange's theorem, relating the order of a subgroup to the order of a group:

For finite groups, this can be restated as:

• Lagrange's theorem: If H is a subgroup of a finite group G, then the order of H divides the order of G.
• If the order of group G is a prime number, G is cyclic.