Elementary group theory

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.

Axioms A1 and A2 follow from the definition of "associative binary operation", and are sometimes omitted, particularly A1.

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 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 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.

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 1.3, 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 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 x⁰ = e. We write x*x as x² ; and in general, x*x*x*...*x (n times) as xⁿ. Similarly, we write x ^-n for (x ^-1)ⁿ.

Theorem: 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>.

If there is a positive integer n such that aⁿ=e, then we say the element a has order n in G. 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.

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References

• Group Theory, W. R. Scott, Dover Publications, ISBN 0-486-65377-3