ALGEBRA MCQs Test 01

3000+ Mathematics all subject MCQs with their Answeers

Algebra mcqs tests 01 consist of 10 most important multiple choice questions. Prepare these questions for better results and also you can prepare definitions of algebra.

Algebra MCQs Test 01

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1. $\dpi{120} \small \forall \,\,\, a \in A$ The R is a reflexive relation $\dpi{120} \small \bigleftrightarow$ $\dpi{120} \small \Leftrightarrow$

$\dpi{120} \small (a , a) \in R$

a=a

$\dpi{120} \small a \in R$

$\dpi{120} \small (a, a) \notin R$

2. A relation is called anti-symmetric if $\dpi{120} \small (a, b) \in R \,\,\ and \,\,\ (b, a) \in R$ implies

a≠b

a=b

$\dpi{120} \small a=b^{-1}$

3. Relation R is symmetric if $\dpi{120} \small a, b\in A \,\,\, and \,\,\,\ (a, b) \in R$ then

a≠b

a=b

A= ab

$\dpi{120} \small (b , a) \in R$

4. Relation on R is transitive if $\dpi{120} \small (a, b) \in R,(b, c) \in R$ then

$\dpi{120} \small (a, c) \in R$

a=c

$\dpi{120} \small (a, c) \notin R$

$\dpi{120} \small a \in R$

5. for a fixed point $\dpi{120} \small c \in R \,\ and \,\ \phi_c=(x,c)$ is known as

Constant function

None of these

1-function

Identity function

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1. Let G be a cyclic group of order 24. Then order of $a^9$ is

2. The set of cube roots of unity is a subgroup of

C-{0}

R†

3. If X and Y are two sets, then X∩(XUY)’=0

X∩Y

4. Any group G van be embedded in a group of bijective mappings of certain sets is a statement of

Cayley’s theorem

Lagrang’s theorem

Isomorphism theorem

Cauchy’s theorem

5. Let An be the set of all even permutations of Sn is a subgroup of Sn. Then order of An is

n+1/2

n!/3

n!/2

6. Let $D_4=\left \{ <a,b>;a^4=b^2=(ab)^2=1) \right \}$ be a dihedral group of order 8. Then which of the following is a subgroup of D4

$\left \{ <a,b>;b^2=1 \right \}$

$\left \{ <a^3,b>;(a^3b)^2=1) \right \}$

$\left \{ <a,b>;(ab)^2=1) \right \}$

$\left \{ <a,b>;a^4=b^2=1) \right \}$

7. Let G be a finite group. Let H be a subgroup of G . Then which of the following divides the order of G

order of H

order of G

All of these

index of H

8. Which of the following is abelian

$S_5$

$S_3$

$S_4$

$S_2$

9. The union of all positive even and all positive odd integers is

Z†

10. The symmetries of rectangle form a

Kleins four group D4

Permutation group of order 3, S3

optic Group

Dihedral group of order 8

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Algebra MCQs Test 07

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1. $\Phi : R^{+}\rightarrow R$ is an isomorphism. then for all $x \in R^{+}$ which of the following is true.

$\Phi(x)=tan(x)$

$\Phi(x)=x^2+1$

$\Phi(x)=log(x)$

$\Phi(x)=x$

2. In a group of even order there at least ______ elements of order 2.

None

3. Which of the following is cyclic group

4. The group Sn is called

symmetric group of degree n

dihidral group of degree n

None of these

polynomial group of degree n

5. If n(U)= 700, n(A)=200, n(B)=300 and n(A∩B)=100 then n(A’∩B’)=?

300

400

600

240

6. Let G be a cyclic group of order 10. The number of subgroups of G is

7. Let G be a cyclic group. Then which of the following is cyclic

subgroup of G

all of these

center of G

homomorphic of G

8. In S4 group of permutation, number of even permutation is

9. Suppose that n(A)=3 and n(B)=6 then what can be minimum number of elements

10. If a group is neither periodic nor torsion free then G is

mixed group

Free group

symmetric Group

infinite group

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Algebra MCQs Test 06

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1. which of the following is even permutation

$\bigl(\begin{smallmatrix} 1 & 2 & 3 & 4\\ 2 & 4 & 1 & 3 \end{smallmatrix}\bigr)$

None

$\bigl(\begin{smallmatrix} 1 & 2 & 3 & 4\\ 2 & 1 & 4 & 3 \end{smallmatrix}\bigr)$

$\bigl(\begin{smallmatrix} 1 & 2 & 3 & 4\\ 2 & 3 & 1 & 4 \end{smallmatrix}\bigr)$

2. If X and Y are two sets s.t n(x)=17, n(Y)=23 and n(X∪Y)=38 then n(X∩Y)=?

3. The group in which every element except the identity element has infinite order is called

locally infinite

all of these

a-periodic

torsion free

4. Let X has n elements. The Set Sn of all permutations of X is a group w.r.t to mappings

composition

multilpication

inverse

addition

5. Let G be an infinite cyclic group . Then the number of generators of G are

6. R+ is a group of non-zero positive real number under multiplication. Then which of the following group under addition is isomorphic to R+

7. Number of non-empty subsets of the set {1,2,3,4}

8. Let G be a cyclic group . Then which of the following cab be order of G.

1000000007

all of these

107

9. Let G be a group and a,b ∈ G then order of $a^{-1}$ =

$bab^{-1}$

$b^{-1}$

10. Let G be a cyclic group of order 17. The number of subgroups of G are

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1. Automorphism and inner automorphism of a group G are

normal

conjugate

abelian

none of these

2. The intersection of any collection of normal subgroups of a group is

Not Abelian

Abelian

Equivalent

Normal

3. Equivalence relation between subgroups of a group is a relation

congugacy

Isomorphic and congugacy

Homomorphic

Isomorphic

4. Aytomorphism group of a finite group is

infinite

normal

abelian

finite

5. Any two conjugate subgroups have same

order and center

center

order

none of these

6. Every subgroup of an abelian group is

Equivalent

Conjugate

Normal

center

7. Every group of order $P^6$ where P is a prime number is

cyclic

abelian

conjugate

normal

8. The set A(G) of all automorphism ofa group is

Not Group

Normal group

None of these

Group

9. Two conjugate subgroups are

normal

Centralizer

Homomorphic

Isomorphic

10. Group obtained by the direct product of sylow- p group is

normal

abelian

center

commutator

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Algebra MCQs Test 04

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1. A homomorphism P: G ⇒G which is bijective is known as

Isomorphism

Monomorphism

Epimorphism

Endomorphism

2. Any group G be embeded in a groyp of a certain set of

Injective mappings

Surjective mappings

aidentity mappings

Bijective mappings

3. Every permutation can be written as

product of transpositions

both a and b

addition of transpositions

Difference of transpositions

4. The homomorphic image Φ(G) of a group G under homomorphic Φ is itself a

Not Group

Group

Abelian Group

None of these

5. If there is a function f:W→A then aet A is said to be

Countable

finite

infinite

None of these

6. Every subgroup of a cyclic group is

Abelian and Cyclic

Cyclic

Abelian

Non cyclic

7. Every permutation of degree n can be written as a product of

Group

cyclic permutations

Permutations

Non cyclic permutationa

8. The center of a finite P- group is

Non.Trivial

Conjugate

Normal

Trivial

9. A homomorphic image of a cyclic group is

Abelian

Non cyclic

Isomorphic

Cyclic

10. Any two cyclic group of same order are

None of these

Isomorphic

meromorphic

Homomorphic

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Algebra MCQs Test 03

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1. If $\Psi: A\rightarrow B$ be a function and for $a \in A,b \in B\,\ ,\Psi(a)\neq \Psi (b)\,\ for \,\ a \neq b$ then function is known as

Onto

Constant

1-1

Identity

2. A homomorphic image of cyclic group is

Cyclic

Non Cyclic

Abelian

Isomorphic

Subgroup G generated by all commutators [u, v] such that u,v∈G then it is known as

Normal Subgroup

Abelian

Commutator and Normal Subgroup

Commutator subgroup

4. Let H be a subgroup of G and for fixed element of G then we define $K=hgh^{-1}=\left \{ghg^{-1}: h\in H \right \}$ then K is

Center

Trivial

Conjugate subgroup of H

Normal

5. Every group of order square of prime number is known as

Normal group

Both Cyclic and Abelian

cyclic group

Abelian Group

6. An endomorphism $\phi :G\rightarrow G$ is said to be automorphism if $\phi$ is

Identity

onto

1-1

Bijective

7. Let H and G be the two groups and H⊆G then

$N_G(X) = H$

$hH=H=Hh$

$N_G(X) \supseteq H$

None of these

8. If H is a normal subgroup of G then

HG=H

H≠G

gH=Hg

H=G

9. Let (Z,+) and (E,+) be the groups of integers and even numbers with mappings F:Z→E s.t f(x)=2x for all x∈ Z then function f is known as

Monomorphism

None of these

Homomorphism

Epimorphism

10. Every group of order prime is

Abelian Group

Simple group

Normal group

Mixed group

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1. If $\small u,v \in G$ and for some $\small x \in G$ then v is known as conjugate of u if

$\small v=u^{-1}x$

$\small v=ux^{-1}$

$\small v=xux^{-1}$

$\small v=ux$

2. Let H be a normal subgroup of G then Quotient Group G/H is represented as

$aHa^{-1}$

$\left \{ aH: a\in G \right \}$

$\left \{ a+H: a\in G \right \}$

$aH^{-1}$

3. If $\small H_1 \,\ and \,\ H_2$ be the subgroups of a group G then $\small H_1\cup H_2$ is a subgroup of G if and only if

$\small H_1=H_2$

$\small H_1\subseteq H_2,H_2\subseteq H_1$

$\small H_1\cap H_2=\Phi$

$\small H_1\neq H_2$

4. Every group whose order is a prime number is necessary

Non-Cyclic

aBELIAN

Non-Abelian

Cyclic

5. In a group G if there are n integers such that $\small a^n=e$ then order of a group is

Finite

Zero

Infinte

abelian

6. The set which is neither finite nor countable is known as

finite

Uncountable

Finite and Uncountable

infinite

7. The Set $\small C_n=[e^{2\pi ki/n} : k={0,1,2,3,...}]$ is a cyclic group of order

8. Two Conjugate elements have

Same Order

No Order

Different Order

None

9. Let A and B subgroups of a group such that A is normal G then normal suproup of B is

$A-B$

$A'\cup B'$

$A\cap B$

$A\cup B$

10. Every group in which each non identity element is of order 2 is

Normal

Abelian

Conjugate

Cyclic

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