Page 121 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 10.3 Notation

There are two main notational conventions for finite abelian groups: ‘+’ additive and ‘.’
multiplicative.

Convention Operation Identity Powers Inverse

Addition x + y 0 nx –x

Multiplication x * y or xy e or 1 x x
n
–1

Generally, the multiplicative notation is the usual notation for groups, while the additive notation
is the usual notation for modules. The additive notation may also be used to emphasize that a
particular group is abelian, whenever both abelian and non-finite abelian groups are considered.

Multiplication Table

To verify that a finite group is abelian, a table (matrix) - known as a Cayley table - can be
constructed in a similar fashion to a multiplication table. If the group is G = {g = e, g , ..., g } under
2
1
n
.
the operation “, the (i, j)’th entry of this table contains the product g g . The group is abelian if
i
j
and only if this table is symmetric about the main diagonal.
.
.
This is true since if the group is abelian, then g g = g g . This implies that the (i, j)’th entry of the
i
i
j
j
table equals the (j, i)’th entry, thus the table is symmetric about the main diagonal.
Examples:
1. For the integers and the operation addition “+”, denoted (Z,+), the operation + combines
any two integers to form a third integer, addition is associative, zero is the additive
identity, every integer n has an additive inverse, –n, and the addition operation is
commutative since m + n = n + m for any two integers m and n.
2. Every cyclic group G is abelian, because if x, y are in G, then xy = a a = a m + n = a n + m = a a =
m n
n m
yx. Thus the integers, Z, form a finite abelian group under addition, as do the integers
modulo n, Z/nZ.
3. Every ring is a finite abelian group with respect to its addition operation. In a commutative
ring the invertible elements, or units, form an abelian multiplicative group. In particular,
the real numbers are a finite abelian group under addition, and the non-zero real numbers
are a finite abelian group under multiplication.
4. Every subgroup of a finite abelian group is normal, so each subgroup gives rise to a quotient
group. Subgroups, quotients, and direct sums of finite abelian groups are again abelian.
In general, matrices, even invertible matrices, do not form a finite abelian group under
multiplication because matrix multiplication is generally not commutative. However, some
groups of matrices are finite abelian groups under matrix multiplication - one example is the
group of 2 x 2 rotation matrices.

Example: Find all finite abelian groups of order 108 (up to isomorphism).

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