Groups-Their Definition and Basic Properties

1 Definition(s)

1.1 Definition of Group and Abelian Group

Let G be a set with a binary operation m (from G × G to G) which we call multiplication. We shall often write ab for m(a, b). We say that G is a group if:
  1. For all a, b, c in G we have a(bc) = (ab)c. This property is called associativity.
  2. There exists an identity element 1 in G such that 1.a = a.1 = a for all a in G. And,
  3. For all a in G there exists an element i(a) such that a.i(a) = i(a).a = 1. The element i(a), which is also denoted by a-1, is called the inverse of a.
A group G is said to be abelian if (in addition to conditions 1-3 above) the following condition is satisfied:
  1. For all a, b in G we have ab = ba. This property is called commutativity.

1.2 Powers: Definition and the power laws

For all g in G and (all) integers n we define the nth­power of g, denoted gn, as follows:
  1. We define g0 = 1 (and g1 = g).
  2. For n > 0, we define gn = gn-1.g. This is a recursive definition.
  3. For n < 0, we define gn = i(g-n).
The following properties of the power map hold:
  1. For all g in G and for all integers m, n we have gmn = (gm)n.
  2. For all g in G and for all integers m, n we have gm+n = gm.gn.
  3. For all g in G and for all integers n we have g-n = i(gn).
  4. For all g, h in G such that g and h commute and for all integers n we have (gh)n = gn.hn.
These properties are referred to as power laws.

1.3 Definition of order (for groups and elements)

The order of G, which we denote by |G| or o(G) is the number of elements that G contains. For infinte (and also finite) groups it is the cardinality of the set G. We have |G| > 1 since Axiom 2 ensures that G cannot be empty.

For g in G, we define o(g), the order of g, to be the least positive (non­zero) n such that gn = 1 when such an n exists, and if such an n does not exist we define o(g) = infinity (or 0).

2 Some Examples

Example 2.1: Trivial Group

Example 2.2: Cyclic group

Example 2.3: Dihedral group

3 Lemmas

Lagrange

Let H be a subgroup of G. Then |H| divides |G|.

The converse of this theorem is false. For example, the alternating group A4, which has order 12, has no subgroup of order 6.

Sylow

Let G be a finite group of order pn.m, where p is prime and m is not divisible by p. We say that a subgroup P is a Sylow p­subgroup just if P has order pn. Sylow's Theorem asserts that Sylow p­subgroups exist (and some other things).

Proof of Sylow's Theorem

Let S denote the set of subsets of G of size pn [where |G| = pn.m as above]. Let G act on these sets [ie S] by right mulitplication, so that the stabiliser of any element of S has order at most pn. Now |S| is the coefficient of pn in (1 + x)pn.m, and using (1 + x)p = 1 + xp (mod p), we get |S| = m (mod p). Since p does not divide m, we get that G has at least one orbit on S whose size is not divisible by p. Then the Orbit-Stabiliser Theorem tells us that the order of the stabiliser of something in that orbit is divisible by pn. Since pn is the maximum order of such a stabiliser, we get a stabiliser of order pn. This stabiliser is a subgroup and so we are done.
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Last updated 23rd August, 1997
John N. Bray