   #copyright

Elementary group theory

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   In mathematics, a group (G,*) is usually defined as:

   G is a set and * is an associative binary operation on G, obeying the
   following rules (or axioms):

          A1. ( Closure) If a and b are in G, then a*b is in G
          A2. ( Associativity) If a, b, and c are in G, then
          (a*b)*c=a*(b*c).
          A3. ( Identity) G contains an element, often denoted e, such
          that for all a in G, a*e=e*a=a. We call this element the
          identity of (G,*). (We will show e is unique later.)
          A4. ( Inverses) If a is in G, then there exists an element b in
          G such that a*b=b*a=e. We call b the inverse of a. (We will show
          b is unique later.)

   Closure and associativity are part of the definition of "associative
   binary operation", and are sometimes omitted, particularly closure.

   Notes:
     * The * is not necessarily multiplication. Addition works just as
       well, as do many less standard operations.
     * When * is a standard operation, we use the standard symbol instead
       (for example, + for addition).
     * When * is addition or any commutative operation (except
       multiplication), the identity is usually denoted by 0 and the
       inverse of a by -a. The operation is always denoted by something
       other than *, often +, to avoid confusion with multiplication.
     * When * is multiplication or any non-commutative operation, the
       identity is usually denoted by 1 and the inverse of a by a^ -1. The
       operation is often omitted, a*b is often written ab.
     * (G,*) is usually pronounced "the group G under *". When affirming
       that it is a group (for example, in a theorem), we say that "G is a
       group under *".
     * The group (G,*) is often referred to as "the group G" or simply
       "G"; but the operation "*" is fundamental to the description of the
       group.

Examples

(R,+) is a group

   The real numbers (R) are a group under addition (+).

          Closure: Clear; adding any two numbers gives another number.
          Associativity: Clear; for any a, b, c in R, (a+b)+c=a+(b+c).
          Identity: 0. For any a in R, a+0=a. (Hence the denotation 0 for
          identity)
          Inverses: For any a in R, -a+a=0. (Hence the denotation -a for
          inverse)

(R,*) is not a group

   The real numbers (R) are NOT a group under multiplication (*).

          Identity: 1.
          Inverses: 0*a=0 for all a in R, so 0 has no inverse.

(R^#,*) is a group

   The real numbers without 0 (R^#) are a group under multiplication (*).

          Closure: Clear; multiplying any two numbers gives another
          number.
          Associativity: Clear; for any a, b, c in R, (a*b)*c=a*(b*c).
          Identity: 1. For any a in R, a*1=a. (Hence the denotation 1 for
          identity)
          Inverses: For any a in R, a^ -1*a=1. (Hence the denotation a^ -1
          for inverse)

Basic theorems

Inverse relations are commutative

   Theorem 1.1: For all a in G, a^ -1*a = e.
     * By expanding a^ -1*a, we get
          + a^ -1*a = a^ -1*a*e (by A3')
          + a^ -1*a*e = a^ -1*a*(a^ -1*(a^ -1)^ -1) (by A4', a^ -1 has an
            inverse denoted (a^ -1)^ -1)
          + a^ -1*a*(a^ -1*(a^ -1)^ -1) = a^ -1*(a*a^ -1)*(a^ -1)^ -1 =
            a^ -1*e*(a^ -1)^ -1 (by associativity and A4')
          + a^ -1*e*(a^ -1)^ -1 = a^ -1*(a^ -1)^ -1 = e (by A3' and A4')
     * Therefore, a^ -1*a = e

Identity relations are commutative

   Theorem 1.2: For all a in G, e*a = a.
     * Expanding e*a,
          + e*a = (a*a^ -1)*a (by A4)
          + (a*a^ -1)*a = a*(a^ -1*a) = a*e (by associativity and the
            previous theorem)
          + a*e = a (by A3)
     * Therefore e*a = a

Latin square property

   Theorem 1.3: For all a,b in G, there exists a unique x in G such that
   a*x = b.
     * Certainly, at least one such x exists, for if we let x = a^ -1*b,
       then x is in G (by A1, closure); and then
          + a*x = a*(a^ -1*b) (substituting for x)
          + a*(a^ -1*b) = (a*a^ -1)*b (associativity A2).
          + (a*a^ -1)*b= e*b = b. (identity A3).
          + Thus an x always exists satisfying a*x = b.
     * To show that this is unique, if a*x=b, then
          + x = e*x
          + e*x = (a^ -1*a)*x
          + (a^ -1*a)*x = a^ -1*(a*x)
          + a^ -1*(a*x) = a^ -1*b
          + Thus, x = a^ -1*b

   Similarly, for all a,b in G, there exists a unique y in G such that y*a
   = b.

The identity is unique

   Theorem 1.4: The identity element of a group (G,*) is unique.
     * a*e = a (by A3)
     * Apply theorem 1.3, with b = a.

   Alternative proof: Suppose that G has two identity elements, e and f
   say. Then e*f = e, by A3', but also e*f = f, by Theorem 1.2. Hence e =
   f.

   As a result, we can speak of the identity element of (G,*) rather than
   an identity element. Where different groups are being discussed and
   compared, often e[G] will be used to identify the identity in (G,*).

Inverses are unique

   Theorem 1.5: The inverse of each element in (G,*) is unique;
   equivalently, for all a in G, a*x = e if and only if x=a^ -1.
     * If x=a^ -1, then a*x = e by A4.
     * Apply theorem 1.3, with b = e.

   Alternative proof: Suppose that an element g of G has two inverses, h
   and k say. Then h = h*e = h*(g*k) = (h*g)*k = e*k = k (equalities
   justified by A3'; A4'; A2; Theorem 1.1; and Theorem 1.2, respectively).

   As a result, we can speak of the inverse of an element x, rather than
   an inverse.

Inverting twice gets you back where you started

   Theorem 1.6: For all a belonging to a group (G,*), (a^ -1)^ -1=a.
     * a^ -1*a = e.
     * Therefore the conclusion follows from theorem 1.4.

The inverse of ab

   Theorem 1.7: For all a,b belonging to a group (G,*),
   (a*b)^ -1=b^ -1*a^ -1.
     * (a*b)*(b^ -1*a^ -1) = a*(b*b^ -1)*a^ -1 = a*e*a^ -1 = a*a^ -1 = e
     * Therefore the conclusion follows from theorem 1.4.

Cancellation

   Theorem 1.8: For all a,x,y, belonging to a group (G,*), if a*x=a*y,
   then x=y; and if x*a=y*a, then x=y.
     * If a*x = a*y then:
          + a^ -1*(a*x) = a^ -1*(a*y)
          + (a^ -1*a)*x = (a^ -1*a)*y
          + e*x = e*y
          + x = y
     * If x*a = y*a then
          + (x*a)*a^ -1 = (y*a)*a^ -1
          + x*(a*a^ -1) = y*(a*a^ -1)
          + x*e = y*e
          + x = y

Repeated use of *

   Theorem 1.9: For every a in a group (G,*), we define

          \begin{matrix}&a*a*a*a*...*a&\mbox{m times}\\
          &*a*a*a*...*a&\mbox{n times}\end{matrix}

   as :

          \begin{matrix} a^m*a^n &=& a^{m+n}\\ &=& a^n*a^m \end{matrix}

   and

          \begin{matrix} (a^n)^m&=& (a^m)^n \end{matrix}

   and

          \begin{matrix} a^{-n}= a^{-1}*a^{-1}*a^{-1}*...*a^{-1} &\mbox{n
          times} \end{matrix}

   However, when the operation is noted +, we note

          \begin{matrix}&a+a+a+a+...+a&\mbox{m times}\\
          &+a+a+a+...+a&\mbox{n times}\end{matrix}

   as :

          \begin{matrix} (n+m)a&=& (m+n)a\\ &=& ma+na\\ &=& na+ma
          \end{matrix}

   and

          \begin{matrix} -na = (-a)+(-a)+(-a)+(-a)+...+(-a) &\mbox{n
          times} \end{matrix}

   Where n,m \in \mathbb{Z} (This generalizes the associativity.)

Groups in which all non-trivial elements have order 2

   Theorem 1.10: A group where all non-trivial elements have order 2 is
   abelian. In other words, if all elements g in a group G satisfy g*g=e,
   then for any 2 elements g, h in G, g*h=h*g.
     * Let g, h be any 2 elements in a group G
     * By A1, g*h is also a member of G
     * Using the given condition, we know (g*h)*(g*h)=e. Now
          + g*(g*h)*(g*h) = g*e
          + g*(g*h)*(g*h)*h = g*e*h
          + (g*g)*(h*g)*(h*h) = (g*e)*h
          + e*(h*g)*e = g*h
          + h*g = g*h
     * Since the group operation commutes, the group is abelian

Definitions

   Given a group (G, *), if the total number of elements in G is finite,
   then the group is called a finite group. The order of a group (G,*) is
   the number of elements in G (for a finite group), or the cardinality of
   the group if G is not finite. The order of a group G is written as |G|
   or (less frequently) o(G).
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   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,*).
     * If h is in H, then h*e[H] = h; since h must also be in G, h*e = h;
       so by theorem 1.4, 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.
     * 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. One handy theorem that covers the case for both
   finite and infinite 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.
     * 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.
     * 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].

   In a group (G,*), define x^0 = e. We write x*x as x² ; and in general,
   x*x*x*...*x (n times) as x^n. Similarly, we write x^ -n for (x^ -1)^n.

   Theorem 2.5: Let a be an element of a group (G,*). Then the set {a^n: 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>.

   If there exists a positive integer n such that a^n=e, then we say the
   element a has order n in G where n is the smallest n. Sometimes this is
   written as "o(a)=n".
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   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.

   Some useful theorems about cosets, stated without proof:

   Theorem: If H is a subgroup of G, and x and y are elements of G, then
   either x*H = y*H, or x*H and y*H have empty intersection.

   Theorem: If H is a subgroup of G, every left (right) coset of H in G
   contains the same number of elements.

   Theorem: If H is a subgroup of G, then G is the disjoint union of the
   left (right) cosets of H.

   Theorem: If H is a subgroup of G, then the number of distinct left
   cosets of H is the same as 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 also allows us to state:

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

   Theorem: If the order of a group G is a prime number, then the group is
   cyclic.
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