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 $\dpi{120} \small \bigleftrightarow$ $\dpi{120} \small \Leftrightarrow$

a=a

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

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

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

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

a=b

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

a≠b

A= ab

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

a=c

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

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

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

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

Identity function

None of these

Constant function

1-function

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

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

a=b

a≠b

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

X∩Y

2. The symmetries of rectangle form a

optic Group

Kleins four group D4

Permutation group of order 3, S3

Dihedral group of order 8

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

Z†

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

C-{0}

R†

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

All of these

order of G

index of H

order of H

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

n!/3

n!/2

n+1/2

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

Cauchy’s theorem

Lagrang’s theorem

Cayley’s theorem

Isomorphism theorem

8. 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

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

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

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

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

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

10. Which of the following is abelian

$S_5$

$S_2$

$S_4$

$S_3$

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Algebra MCQs Test 07

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1. The group Sn is called

symmetric group of degree n

polynomial group of degree n

None of these

dihidral group of degree n

2. Which of the following is cyclic group

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

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

homomorphic of G

center of G

all of these

subgroup of G

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

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

mixed group

Free group

symmetric Group

infinite group

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

None

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

600

300

240

400

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

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

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

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

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

$\Phi(x)=x$

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Algebra MCQs Test 06

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

107

1000000007

all of these

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

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

a-periodic

torsion free

locally infinite

all of these

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

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

6. which of the following is even permutation

None

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

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

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

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

multilpication

addition

inverse

composition

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

$b^{-1}$

$bab^{-1}$

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

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

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1. Any two conjugate subgroups have same

order and center

none of these

center

order

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

Isomorphic and congugacy

Homomorphic

Isomorphic

congugacy

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

Not Abelian

Normal

Equivalent

Abelian

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

Not Group

Group

Normal group

None of these

5. Two conjugate subgroups are

Homomorphic

normal

Centralizer

Isomorphic

6. Aytomorphism group of a finite group is

normal

infinite

finite

abelian

7. Automorphism and inner automorphism of a group G are

conjugate

none of these

normal

abelian

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

normal

abelian

conjugate

cyclic

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

abelian

normal

commutator

center

10. Every subgroup of an abelian group is

center

Equivalent

Normal

Conjugate

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Algebra MCQs Test 04

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

Endomorphism

Isomorphism

Epimorphism

Monomorphism

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

finite

infinite

None of these

Countable

3. Every subgroup of a cyclic group is

Non cyclic

Abelian and Cyclic

Cyclic

Abelian

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

cyclic permutations

Group

Permutations

Non cyclic permutationa

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

aidentity mappings

Surjective mappings

Injective mappings

Bijective mappings

6. Every permutation can be written as

addition of transpositions

Difference of transpositions

both a and b

product of transpositions

7. The center of a finite P- group is

Conjugate

Non.Trivial

Normal

Trivial

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

Not Group

Abelian Group

None of these

Group

9. Any two cyclic group of same order are

Homomorphic

meromorphic

None of these

Isomorphic

10. A homomorphic image of a cyclic group is

Non cyclic

Cyclic

Isomorphic

Abelian

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Algebra MCQs Test 03

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1. If H is a normal subgroup of G then

HG=H

gH=Hg

H=G

H≠G

2. Every group of order prime is

Abelian Group

Mixed group

Simple group

Normal group

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

cyclic group

Abelian Group

Normal group

Both Cyclic and Abelian

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

$N_G(X) = H$

$N_G(X) \supseteq H$

$hH=H=Hh$

None of these

5. 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

Epimorphism

Homomorphism

None of these

Monomorphism

6. A homomorphic image of cyclic group is

Non Cyclic

Abelian

Cyclic

Isomorphic

7. 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

Normal

Trivial

Center

Conjugate subgroup of H

8. 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

Constant

Onto

1-1

Identity

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

Identity

onto

Bijective

1-1

10.

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

Abelian

Normal Subgroup

Commutator and Normal Subgroup

Commutator subgroup

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

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

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

$aH^{-1}$

$aHa^{-1}$

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

Zero

Infinte

abelian

Finite

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

aBELIAN

Non-Cyclic

Cyclic

Non-Abelian

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

$A\cup B$

$A-B$

$A\cap B$

$A'\cup B'$

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

finite

Uncountable

infinite

Finite and Uncountable

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

$\small v=ux$

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

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

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

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

Abelian

Cyclic

Conjugate

Normal

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

9. 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

$\small H_1=H_2$

$\small H_1\cap H_2=\Phi$

$\small H_1\neq H_2$

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

10. Two Conjugate elements have

None

Same Order

No Order

Different Order

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WATU 27]

PAKMATH Knowledge Learning Point

ALGEBRA MCQs Test 01

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