Groups and Rings

If G is a group of order p, where p is a prime number. Then the number of subgroups of G is

• 1

• 2

• p-1

• p

The set of all real numbers under the usual multiplication operation is not a group since

• multiplication is not a binary operation

• multiplication is not associative

• identity element does not exist

• zero has no inverse

If (G, .) is a group such that a²=e , for all a∈G, then G is

• semigroup

• abelian group

• non-abelian group

• none of these

In the group (G, .), the value of (a^-1b)^-1 is

• ab^-1

• b^-1a

• a^-1b

• ba^-1

Every subgroup of a cyclic group is

• cyclic and normal

• cyclic but not normal

• normal but not cyclic

• neither cyclic nor normal

There may be a group in which the cancellation law fails

• True

• False

Every subset of every group is a subgroup under the induced operation

• True

• False

In every cyclic group, every element is a generator

• True

• False

The set of integers Z with the binary operation " ∗" defined as a∗b=a+b+1 for a, b∈Z, is a group. The inverse of a is

• 0

• -2

• a-2

• -a-2

A cyclic group has a unique generator

• True

• False