Abstract Algebra Answers

Consider S_n for n ≥ 2 and let σ be a fixed odd permutation. Show that every odd permutation in S_n is a product of σ and a permutation in A_n

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Prove that for every subgroup H of S_n , either all of the permutations in H are even, or exactly half are even.

 a) Using Cayley’s theorem, find the permutation group to which a cyclic group of 

order 12 is isomorphic. (4) 

 b) Let τ be a fixed odd permutation in .

S10 Show that every odd permutation in S is 

10

a product of τ and some permutation in .

A10 (2) 

 c) List two distinct cosets of < r > in ,

D10 where r is a reflection in .

D10 (2) 

 d) Give the smallest n ∈ N for which An is non-abelian. Justify your answer. 

a) Prove that every non-trivial subgroup of a cyclic group has finite index. Hence 

prove that (Q, +) is not cyclic. (7) 

 b) Let G be an infinite group such that for any non-trivial subgroup H of 

G, G : H < ∞. Then prove that 

 i) 

H ≤ G ⇒ H = {e} or H is infinite; 

 ii) If g ∈G, g ≠ e, then o(g) is infinite. (5) 

 c) Prove that a cyclic group with only one generator can have at most 2 elements. (3) 

Which of the following statements are true? Give reasons for your answers. (10)

i) If a group G is isomorphic to one of its proper subgroups, then G = Z.

ii) If x and y are elements of a non-abelian group (G, ∗) such that x ∗ y = y ∗ x,

then x = e or y = e, where e is the identity of G with respect to .

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iii) There exists a unique non-abelian group of prime order.

iv) If (a, b)∈A× A, where A is a group, then o((a, b)) = o(a)o(b).

v) If H and K are normal subgroups of a group G, then hk = kh ∀ h ∈H, k ∈K.

verify whether (z4,+4) is a group z4 is the set of integers modulo 4 and +4 is addition modulo 4

Find the zero divisors of R × Z₂ × Z₄

Cancellation law may not hold in an arbitrary ring.

Let 𝑛 n be a positive integer that is at least 3 . Show that 𝑈(2^𝑛) has at least three elements of order 2. Prove using Bezout's theorem

Which of the following mapping are group homomorphisms? In case they are group homomorphisms find the kernel.

(i) f:R - {0} - R {0} under multiplication defined by f{0}=|x|,

(ii) f:z - R under addition defined by f(x) = x