This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Graph’s Matrices”.

1. A direct product of a group G possess which of the following characteristics?

a) a multiplication of subgroups of G

b) a factorization via subgroups of G

c) a superset of subgroups of G

d) a maximal power set of subgroups

View Answer

Explanation: A direct product of a group G is a factorization via subgroups of G when the intersection is nontrivial, say X and Y, such that G = XY, X intersect Y = 1, and [X, Y]=1 and X, Y are normal in G.

2. In invariant algebra, some generators of group G1 that goes either into itself or zero under ______ with any other element of the algebra.

a) commutation

b) permutation

c) combination

d) lattice

View Answer

Explanation: Some generators of group G1 in group theory which goes either into itself or zero under commutation with any other element of the whole algebra is called invariant subalgebra.

3. Which of the following can be embedded in an algebraically closed group?

a) infinite group

b) stargraph

c) a countable group

d) a semilattice

View Answer

Explanation: We know that any countable group can always be embedded in an algebraically closed group.

4. Which of the following is the set of m×m invertible matrices?

a) a permutation group of degree m^{2}

b) a general linear group of degree m

c) a sublattice group of degree m

d) a isomorphic graph of m nodes

View Answer

Explanation: The general linear group of degree m is the set of m×m invertible matrices, consists of a general linear group of degree m having the ordinary matrix multiplication operation.

5. If any group is a manifold what is the dimension of that group?

a) same as manifold

b) same as vector space

c) infinite

d) finite

View Answer

Explanation: If a group is a (topological) manifold, then the dimension of a group will be the dimension of this manifold. A linear representation F of a group G

_{1}on a vector space V’ has the dimension of V’.

6. A Latin square graph is a representation of a _______

a) quasi group

b) homomorphic group

c) semigroup

d) subgroup

View Answer

Explanation: We know that any group is a representation of a graph. Now, a Quasi Group can be represented by a Latin Square matrix or by a Latin Square graph.

7. There exists _______ between group homology and group cohomology of a finite group.

a) homomorphism

b) isomorphism

c) automorphism

d) semilattice structure

View Answer

Explanation: We know that there exists an isomorphism between group homology and group cohomology of finite group. Let S’ denote the set of all integers, and let G’ be a finite cyclic Group and for every S then G’-module N, we have S’S’

_{n}(G’, A) is isomorphic to S’

_{n+1}(G’, A).

8. In basic ring theory, any ring R1 may be embedded in its own ________

a) semilattice

b) endomorphism ring

c) homomorphic ring

d) subgroup

View Answer

Explanation: We know that in basic ring theory, any ring R with its identity can be embedded in its own endomorphism ring and this is one of the most important characterization of rings. The endomorphism ring can contain a copy of its ring.

9. In Modern particle physics there must exist ______________

a) group theory

b) graph theory

c) lattice structure

d) invariant semigroup

View Answer

Explanation: Modern particle physics exists with group theory. Group theory can predict the existence of many elementary particles. Depending on different symmetries, the structure and behaviour of molecules and crystals can be defined.

10. For any graph say G, Cayley graph is ______________

a) canonial

b) not canonical

c) isomorphic

d) homomorphic

View Answer

Explanation: A different Cayley graph will be given for each choice of a generating set. Hence, the Cayley graph is not canonical.

**Sanfoundry Global Education & Learning Series – Discrete Mathematics.**

To practice all areas of Discrete Mathematics, __here is complete set of 1000+ Multiple Choice Questions and Answers__.

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