Page 128 - DMTH403_ABSTRACT_ALGEBRA

Unit 10: Finite Abelian Groups

2. In a finite abelian group, each element is in a conjugacy class by itself and the character Notes
table involve powers of a single element known as a ..................
(a) group generator (b) group connector
(c) group and subgroup (d) normal group element
3. In mathematica, the function finite abelian group {n , n , .... } represents .................. product
1
2
of the cyclic group of degree n n ..................
2
1
(a) direct (b) indirect
(c) single (d) external
4. In commutative ring .................. the elements, or unit, from an abelian multiplication
groups.
(a) inversible (b) vertible

(c) direct (d) finite
5. Every subgroup of a finite abelian group is normal, so each subgroup gives rest to a
.................. group.

(a) cyclic (b) permutation
(c) quotient (d) multiplicative

10.4 Summary

A finite abelian group is a set, A, together with an operation “•” that combines any two

elements a and b to form another element denoted a • b. The symbol “•” is a general
placeholder for a concretely given operation. To qualify as a finite abelian group, the set
and operation, (A, •), must satisfy five requirements known as the finite Abelian group
axioms.
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.
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.
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 =
n m
m n

yx. Thus the integers, Z, form a finite abelian group under addition, as do the integers
modulo n, Z/nZ.
10.5 Keywords

Finite Abelian Group: A finite abelian group is a set, A, together with an operation “•” that
combines any two elements a and b to form another element denoted a • b.
Multiplication: The multiplicative notation is the usual notation for groups, while the additive
notation is the usual notation for modules.

Cyclic Group: Every cyclic group G is abelian, because if x, y are in G, then xy = a a = a m + n =
m n
a n + m = a a = yx.
n m

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