Assignment #2, 06.12.13: Group Theory II

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(1)

Suppose that G is a group and that the set {x ∈ G | |x| = 2} has exactly one element. Show that G is abelian.

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2

You are given that G is group of order 24 which is not isomorphic to S4 . Show that one of its Sylow subgroups is normal.

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3

Determine the number of pairwise non-isomorphic groups of order pq, where p and q are prime.

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4

Let ϕ : G → H be a homomorphism of groups. Let G # and H # denote the set of conjugacy classes in G and H , respectively.

(a) Show that ϕ induces a map ϕ# : G# → H#.

(b) Show that if ϕ# is injective, so is ϕ.

(c) Show that if ϕ# is surjective, and H is finite, then ϕ is surjective. (Hint: one of the problems from PS1 might be useful here...)

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5

Let G be a group of order 56 with a normal 2-Sylow subgroup Q, and let P be a 7-Sylow subgroup of G. Show that G∼= P × Q or Q ∼= Z × Z × Z.

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6

Let G be a group and H a subgroup of G with finite index. Show that there exists a normal subgroup N of G of finite index with N ⊂ H.

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7

Let G be a finite group and P a p-Sylow subgroup of G for some prime p. (You may assume that p divides |G|. I haven’t had enough coffee to think about the implications of the vacuous case...)

(a) Assume p=2 and P is cyclic. Show that the normalizer and centralizer of P coincide.

(b) Show that this may not hold if p = 2 but P is not cyclic.

(c) Show that the first statement does not hold regardless of cyclicity if p ̸= 2.

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8

Let G be a finite group and ϕ : G → G a homomorphism. Show that ϕ(P ) is a subgroup of P whenever P is a normal Sylow subgroup.

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9

(a) Find all simple groups of order 101.

(b) Find all simple groups of order 102.

(c) Find all groups of order 175.

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10

Let p and q be primes such that p divides q − 1.

(a) Show that there exists a group G with generators x and y and relations xp 2 = 1, yq = 1, xyx−1 = ya , where a is an integer not congruent to 1 mod q, but ap ∼= 1(mod q).

(b) Prove that the Sylow q-subgroup Sq ⊂ G is normal.

(c) Prove that G/Sq is cyclic; and deduce that G has a unique subgroup H of order pq.

(d) Prove that H is cyclic.

(e) Prove that any subgroup of G with order p is contained in H , hence is generated by xp and is contained in the center of G.

(f ) Prove that the center of G is the unique subgroup of G having order p.

(g) Prove that every proper subgroup of G is cyclic.

(h) For each positive divisor d of p2 q, determine the number of elements of G having order d. 2

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