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 $+,0,-a$, 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 $\{G,+\}$ 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:

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

This establishes A4 (and hence A4”).

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

Proof.

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

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

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

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

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

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

As a result, we can speak of the inverse of an element $a$, 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)

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

(3) If $x\perp a=y\perp a$ 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 $n,m\in \mathbb {Z}$:

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

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 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:

• 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:

• 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.

See also

• group theory
• abelian groups
• Glossary of group theory
• List of group theory topics

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