Group Theory Practice Questions

A. 1638

B. 2100

C. 2640

D. None of the above

A. 9

B. 8

C. 7

D. 6

A. $\left(Z, *\right)$ is a monoid

B. $\left(Z, *\right)$ is an Abelian group

C. $\left(Z, *\right)$ is a group

D. None of the above

A. The set of rational numbers is an abelian group under addition

B. The set of integers in an abelian group under addition

C. The set of rational numbers form an abelian group under multiplication

D. The set of real numbers excluding zero is an abelian group under multiplication

A. The set of all bijective functions on a finite set forms a group under function composition.

B. The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod p, where p is a prime number.

C. The set of all strings over a finite alphabet forms a group under concatenation.

D. A subset $S \neq \emptyset$ of G is a subgroup of the group $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$ .

A. A is closed under * but $\langle A, *\rangle$ is not a semigroup.

B. $\langle A, *\rangle$ is a semigroup but not a monoid.

C. $\langle A, *\rangle$ is a monoid but not a group.

D. $\langle A, *\rangle$ is a a group but not an abelian group.

A. $g=g^{-1} \text{ for every } g \in G$

B. $g=g^2 \text{ for every }g \in G$

C. $(goh)^2 = g^2oh^2 \text{ for every } g, h \in G$

D. G is of finite order