Abstract Algebra

True or false: If * is a binary operation on a set G, then a*a=a for all a in G

• True

• False

True or false: Any associative operation is commutative

• True

• False

True or False: Every binary operation on a single element set is both commutative and associative

• True

• False

True or false: Every subgroup of an abelian group is abelian

• True

• False

Every subgroup of a non-abelian group is non abelian

• True

• False

Which of the following is a group under addition?

• The set of all positive integers

• The set of all odd integers

• The set of all negative integers

• The set of all even integers

Which of the following is a group under multiplication?

• The set of integers

• The set of all rational numbers

• The set of all real numbers

• The set of all nonzero rational numbers

The identity element of (nZ,+) is

• 1

• n

• 0

• no identity element

True or false: There may be some groups in which cancellation laws fail

• True

• False

The left cancellation law ab=ac implies b=c in a group is true because

• We can divide both equation by a and get the result

• a is not equal to 0

• left side can be operated with inverse of a

• a can be cancelled from the left