Elementary group theory

In mathematics and abstract algebra, a group is the algebraic structure $\{G,\perp\}$ , where $G$ is a non-empty set and $\perp$ denotes a binary operation $\perp:G\times{}G\rightarrow{}G,$ called the group operation. The notation $\perp(x,y)$ is normally shortened to the infix notation $x\perp{}y$ , or even to $xy$ .

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

A1, Closure. $a\perp{}b\in{}G$ . This axiom is often omitted because a binary operation is closed by definition.
A2, Associativity. $(a\perp{}b)\perp{}c=a\perp(b\perp{}c)$ .
A3, Identity. There exists an identity (or neutral) element $e\in{}G$ such that $a\perp{}e=e\perp{}a=a$ . The identity of $G$ is unique by Theorem 1.4 below.
A4, Inverse. For each $a\in{}G$ , there exists an inverse element $x\in{}G$ such that $a\perp{}x=x\perp{}a=e$ . The inverse of $a$ is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule:

A5, Commutativity. $a\perp{}b=b\perp{}a$ .

Notation

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

The group operation $\perp$ can be interpreted in a great many ways. The generic notation for the group operation, identity element, and inverse of $a$ are $\perp,e,a',$ 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:

Additively by replacing the generic notation by $%2B,0,-a$ , with "+" being infix. Additive notation is typically used when numerical addition or a commutative operation other than multiplication interprets the group operation;
Multiplicatively by replacing the generic notation by $*,1,a^{-1}$ . Infix "*" is often replaced by simple concatenation, as in standard algebra. Multiplicative notation is typically used when numerical multiplication or a noncommutative operation interprets the group operation.

Other notations are of course possible.

Examples

Arithmetic

Take $G:=\mathbb{Z}$ or $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$ , then $\{G,%2B\}$ is an abelian group.
Take $G:=\mathbb{Q}\setminus\{0\}$ or $\mathbb{R}\setminus\{0\}$ or $\mathbb{C}\setminus\{0\}$ , then $\{G,*\}$ is an abelian group.

Function composition

Let $E$ be an arbitrary set, and let $G$ be the set of all bijective functions from $E$ to $E$ . Let function composition, notated by infix $\circ$ , interpret the group operation. Then $\{G,\circ\}$ is a group whose identity element is $id_E:E\rightarrow{}E, x\mapsto{}x.$ The group inverse of an arbitrary group element $f\in{}G$ is the function inverse $f^{-1}.$

Alternative Axioms

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

A3’, left neutral. There exists an $e\in{}G$ such that for all $a\in{}G$ , $e\perp{}a=a$ .
A4’, left inverse. For each $a\in{}G$ , there exists an element $x\in{}G$ such that $x\perp{}a=e$ .

or by the pair:

A3”, right neutral. There exists an $e\in{}G$ such that for all $a\in{}G$ , $a\perp{}e=a$ .
A4”, right inverse. For each $a\in{}G$ , there exists an element $x\in{}G$ such that $a\perp{}x=e$ .

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 $e,$ and for any given $a\in{}G,$ then A4’ says there exists an $x$ such that $x\perp{}a=e$ .

Theorem 1.2: $a \perp x = e.$

Proof. Let $y\in{}G$ be an inverse of $a\perp{}x.$ Then:

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

This establishes A4 (and hence A4”).

Theorem 1.2a: $a \perp e = a.$

Proof.

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

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 $\{G,\perp\}$ is unique.

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

$\begin{align} e & = & e \perp f &\quad (A3'') \\ & = & f &\quad (A3') \\ \end{align}$

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

Inverses are unique

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

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

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

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

Inverting twice takes you back to where you started

Theorem 1.6: For all elements $a$ in a group $\{G,\perp\},$ $(a')' = a$ .

Proof. $a'\perp{}a = e$ and $(a')'\perp{}a' = e$ are both true by A4. Therefore both $a$ and $(a')'$ are inverses of $a'.$ By Theorem 1.5, $a = (a')'.$

Equivalently, inverting is an involution.

Inverse of ab

Theorem 1.7: For all elements $a$ and $b$ in group $\{G,\perp\}$ , $(a\perp{}b)' = b' \perp a'$ .

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

Cancellation

Theorem 1.8: For all elements $a,x,y$ in a group $\{G,\perp\}$ , then $x = y \Leftrightarrow a \perp x = a \perp y \Leftrightarrow x \perp a = y \perp a$ .

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

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

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

Latin square property

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

Proof.
Existence: If we let $x�:= a' \perp b$ , then $a \perp (a' \perp b) = (a \perp a') \perp b = e \perp b = b$ .
Unicity: Suppose $x$ satisfies $a \perp x = b$ , then by Theorem 1.8, $a' \perp (a \perp x) = a' \perp b \Leftrightarrow x = a' \perp b$ .

Powers

For $n \in \mathbb{Z}$ and $a$ in group $\{G,\perp\}$ we define:

$a ^ n�:= \begin{cases} \underbrace{a\perp{}a\perp\cdots\perp{}a}_{n\ \text{times}}, & \mbox{if }n > 0 \\ e, & \mbox{if }n = 0 \\ \underbrace{a'\perp{}a'\perp\cdots\perp{}a'}_{-n\ \text{times}}, & \mbox{if }n < 0 \end{cases}$

Theorem 1.9: For all $a$ in group $\{G,\perp\}$ and $n, m \in \mathbb{Z}$ :

$\begin{matrix} a^m\perp{}a^n &=& a^{m%2Bn}\\ (a^m)^n &=& a^{m*n} \end{matrix}$

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 e_H in H is identical to the identity e in (G,*).

Proof. If h is in H, then h*e_H = h; since h must also be in G, h*e = h; so by theorem 1.8, e_H = 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 {H_i} be a set of subgroups of G, and let K = ∩{H_i}. e is a member of every H_i 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 H_i.
By the previous theorem, h*k^-1 is in H_i
Therefore, h*k^-1 is in ∩{H_i}.

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

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:

And 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:

Lagrange's theorem: If H is a subgroup of G, then |G| = |H|*[G:H].

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.

References

Jordan, C. R and D.A. Groups. Newnes (Elsevier), ISBN 0-340-61045-X
Scott, W R. Group Theory. Dover Publications, ISBN 0-486-65377-3