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

a≠b

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

a=b

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

a=c

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

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

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

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

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

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

a=a

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

4. 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

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

None of these

Identity function

Constant function

1-function

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1. The symmetries of rectangle form a

optic Group

Dihedral group of order 8

Permutation group of order 3, S3

Kleins four group D4

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

Isomorphism theorem

Cauchy’s theorem

Lagrang’s theorem

Cayley’s theorem

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

X∩Y

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

R†

C-{0}

5. 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

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

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

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

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

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

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 G

index of H

All of these

order of H

8. 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!/2

n!/3

9. Which of the following is abelian

$S_2$

$S_5$

$S_3$

$S_4$

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

Z†

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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.

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

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

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

$\Phi(x)=x$

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

3. Which of the following is cyclic group

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

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

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

240

400

600

300

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

homomorphic of G

subgroup of G

all of these

center of G

8. The group Sn is called

None of these

dihidral group of degree n

polynomial group of degree n

symmetric group of degree n

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

None

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

symmetric Group

mixed group

infinite group

Free group

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Algebra MCQs Test 06

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

composition

multilpication

addition

inverse

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

all of these

107

1000000007

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

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

5. which of the following is even permutation

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

None

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

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

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

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

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

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

$b^{-1}$

$bab^{-1}$

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

torsion free

locally infinite

all of these

a-periodic

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1. Aytomorphism group of a finite group is

infinite

normal

finite

abelian

2. Every subgroup of an abelian group is

Equivalent

Conjugate

Normal

center

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

Isomorphic and congugacy

Homomorphic

congugacy

Isomorphic

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

Not Abelian

Normal

Equivalent

Abelian

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

cyclic

abelian

normal

conjugate

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

commutator

normal

abelian

center

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

None of these

Normal group

Group

Not Group

8. Automorphism and inner automorphism of a group G are

none of these

normal

conjugate

abelian

9. Any two conjugate subgroups have same

none of these

order

center

order and center

10. Two conjugate subgroups are

Homomorphic

normal

Centralizer

Isomorphic

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Algebra MCQs Test 04

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1. Every subgroup of a cyclic group is

Non cyclic

Abelian

Cyclic

Abelian and Cyclic

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

Group

Not Group

None of these

Abelian Group

3. The center of a finite P- group is

Non.Trivial

Trivial

Normal

Conjugate

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

Endomorphism

Epimorphism

Monomorphism

Isomorphism

5. Any two cyclic group of same order are

Homomorphic

None of these

meromorphic

Isomorphic

6. A homomorphic image of a cyclic group is

Isomorphic

Cyclic

Abelian

Non cyclic

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

infinite

Countable

finite

None of these

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

Group

Non cyclic permutationa

cyclic permutations

Permutations

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

Bijective mappings

aidentity mappings

Injective mappings

Surjective mappings

10. Every permutation can be written as

both a and b

addition of transpositions

Difference of transpositions

product of transpositions

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Algebra MCQs Test 03

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

Commutator and Normal Subgroup

Abelian

Commutator subgroup

Normal Subgroup

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

cyclic group

Both Cyclic and Abelian

Abelian Group

Normal group

3. Every group of order prime is

Abelian Group

Normal group

Simple group

Mixed group

4. 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

Monomorphism

Homomorphism

None of these

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

Bijective

1-1

onto

Identity

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

$N_G(X) \supseteq H$

None of these

$hH=H=Hh$

$N_G(X) = H$

7. If H is a normal subgroup of G then

H=G

H≠G

gH=Hg

HG=H

8. 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

Conjugate subgroup of H

Trivial

Normal

Center

9. 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

Identity

1-1

10. A homomorphic image of cyclic group is

Isomorphic

Abelian

Non Cyclic

Cyclic

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1. Two Conjugate elements have

Different Order

Same Order

None

No Order

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

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

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

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

$\small v=ux$

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

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

Cyclic

Normal

Conjugate

Abelian

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

abelian

Finite

Zero

Infinte

6. 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

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

$\small H_1=H_2$

$\small H_1\neq H_2$

$\small H_1\cap H_2=\Phi$

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

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

$aHa^{-1}$

$aH^{-1}$

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

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

Cyclic

Non-Cyclic

aBELIAN

Non-Abelian

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

$A\cup B$

$A\cap B$

$A'\cup B'$

$A-B$

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

Uncountable

Finite and Uncountable

finite

infinite

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WATU 27]

PAKMATH Knowledge Learning Point

ALGEBRA MCQs Test 01

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