First Theorems about Groups
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. (G,*) has closure. That is, if a and b are in G, then a*b is in G
- A2. The operation * is associative, that is, if a, b, and c are in G, then (a*b)*c=a*(b*c).
- A3. G contains an identity element, often denoted e, such that for all a in G, e*a=a*e=a.
- A4. Every element in (G,*) has an inverse; if a is in G, then there exists an element b in G such that a*b=b*a=e.
Where no danger of confusion is possible, the group (G,*) will simply be referred to as "the group G"; but it is important to remember that the operation "*" is fundamental to the description of the group. For example, in the real numbers, we can speak of both the group (R,+), which is the additive group of reals with identity 0; and the group (R^{#}, *), which is the multiplicative group of the reals (excluding 0), which has identity 1.
We can state simpler versions of A3 and A4:
- A3'. G contains an identity element, often denoted e, such that for all a in G, a*e=a.
- A4'. Every element in (G,*) has an inverse; for all a in G, there exists an element in G, denoted a^{ -1}, such that a*a^{ -1} = e.
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
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
The following theorem demonstrates a fundamental property enjoyed by groups, which other more general structures (such as semigroups) lack:
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
- a*e = a (by A3)
- Apply theorem 1.3, with b = a.
Theorem 1.4: 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.
Theorem 1.5: For all a belonging to a group (G,*), (a^{ -1})^{ -1}=a.
- a^{ -1}*a = e.
- Therefore the conclusion follows from theorem 1.4.
- (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.
Theorem 1.7: 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
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).
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 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 1.3, e_{H} = e.
- 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 1.3, k = h^{ -1}.
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.
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}.
Theorem: 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 is a positive integer n such that a^{n}=e, then we say the element a has order n in G. Sometimes this is written as "o(a)=n.
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.
References
- Group Theory, W. R. Scott, Dover Publications, ISBN 0-486-65377-3