Math Chapter Wise

EnglishPGT 2013

Q221.If a and b are two elements of a group G such that a4 = e and bab–1 = a2, then

Aa = b Ba = e Cb = e DNone of these

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EnglishPGT 2013

Q222.Let G be any group 'a' be an elements of order 12. Then what will be the degree of a8.

A3 B4 C12 D24

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EnglishPGT 2013

Q223.Let x, y be any elements of a group (G, o). then (x o y)–1 will be equal to

Ax–1/o y–1 Bx–1/oy Cxoy–1 Dy–1/ox–1

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EnglishPGT 2013

Q224.The set of all integers

ABoth are additive and multiplicative groups BMultiplicative but not additive group CIn additive group but not a Multiplicative group DNeither additive group not multiplicative group

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EnglishPGT 2010

Q225.An operation * is defined on the set of real numbers by a*b=1+a+ab, then the operation * is

Acommutative but not associative Bassociative but not commutative Cneither commutative nor associative Dboth commutative and associative

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EnglishPGT 2010

Q226.Which of the following is incorrect?

AEvery cyclic group is Abelian group BEvery Abelian group is cyclic group CSubgroup of cyclic group is cyclic group DEvery group of prime order is Abelian

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EnglishPGT 2010

Q227.Which of the following structures is not group?

A2 ({1,, },.) ω B({1, 1},.) C({1, 1},) − + D({1, 1,, },.) − − i i

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EnglishPGT 2010

Q228.If f: G→ → →G' is a homomorphism of a group G into a Group G' with kernel k, then k is

ANot a subgroup of G Bk = G Ck is normal subgroup of G DNone of these

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EnglishPGT 2009

Q229.Which of the following statements is incorrect?

AA gruoup of prime order is cyclic BEvery cyclic group is Abelian group CEvery Abelian group is cyclic DEvery group of composite order possesses proper subgroup

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EnglishPGT 2009

Q230.From following sets, the group under the indicated operations is

AG = {1, 3, 7, 9}; X10 BG = {0, 1, 2, 3, 4, 5}; X6 CG = Z; m o n = m –n for m, n ∈Z DG = {A: A is 2 × 2 matrix over positive real numbers}: matrix multiplication.

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EnglishPGT 2009

Q231.The coset decomposition of multiplicative group G={1,–1,i,–i} with respect to its subgroup H = {1, –1} is

A{1, –1} and {i,–i } B{1, –1} and {1, –i} C{1, –1} and {1, i } D{1, –1} and {–1, i }

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EnglishPGT 2005

Q232.Which of the following is a group?

A(Z, *) where a*b = ab ∀a, b∈Z B(Z, *) where a*b = a–b ∀a, b∈Z C(R, *) where a*b = ab ∀a, b∈R+ D(Z, *) where a*b = ab ∀a, b∈Z

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EnglishPGT 2005

Q233.Let H be a subgroup of a group G. In which of the following H is not normal?

A{g∈G: gHg–1 = H}≠G BIndex of H in G is 2 CProduct of two right cosets of H is a right coset DProduct of two left cosets of H is a left coset.

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EnglishPGT 2005

Q234.Let φ be a homomorphism of a group G into another group G' and e, e' be their identity elements. Which of the following is not correct?

Aφ(x,y)=φ(x)φ(y) Bφ(x)=φ(y) ⇒ x = y Cφ(e) = e' Dφ(x–1)–1 = (φ(x))

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EnglishPGT 2005

Q235.If in a group G, O(a) = m and O(b) = n where a, b ∈G, then which of the following statements is correct?

AO(ab) = m BO(ab) = n CO(ab)=mn DNone of these

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EnglishPGT 2005

Q236.Which of the following statements is correct?

AOrder of the subset of a group always divides order of the group BCyclic groups have no subgroups COrder of a coset always divides order of the group DOrder of an element of a group never divides the order of the group

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EnglishRajasthan TGT 2011

Q237.The identify element in the set Z of integers under the binary operation * Defined by a b a b 1, ∗ = + + ∗ = + + ∗ = + + ∗ = + + ab z ∀ ∈ ∀ ∈ is

A0 B-1 C1 D2

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EnglishRajasthan TGT 2011

Q238.Let G be a group such that order of G is 13, then G is

AAn abelian group BA cyclic group CAbelian and cyclic both DNeither abelian nor cyclic

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EnglishRajasthan TGT 2011

Q239.If H is a subgroup of a group G then which one of the following statements is incorrect/ Ùeefo H

AHa H = if and only if a H ∈ BHa Hb = if and only if 1 ba H − ∈ COrder of H divides order of G D1 H H − =

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EnglishRajasthan TGT 2011

Q240.Let G=(Z,+) be the additive group of integers and let H=3Z={3x x Z} ∈ be a subgroup of G, then order of the quotient group G/H is

A2 B3 C4 DInfinite

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