Problem 1: Let H be a subgroup of group (G, °).a.) Show that NG(H) / CG(H) is isomorphic to a...

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Problem 1: Let H be a subgroup of group (G, °).a.) Show that NG(H) / CG(H) is isomorphic to a subgroup of Aut(H).b.) Show that CG(H) = NG(H) if |H| = 2.Problem 2: Let G be a group of order |G| = 992 = 25 · 31.a.) Show that at least one of its Sylow p-subgroups is normal.b.) If G is Abelian, find all prime-power decompositions and invariant decompositions.Problem 3: Prove that if G / Z(G) is cyclic, then G is Abelian.Problem 4: Let G be a group of order 99.a.) Prove that there exists a subgroup of H of order 3. (Should be a one-sentence proof.)b.) Prove there is a unique subgroup K of G such that K / H ˜ Z3 and G / K ˜ Z11.Problem 5: Characterize those integers n such that the only Abelian groups of order n are cyclic.Problem 6: Express U(165) as an internal direct product of proper subgroups in 4 different ways.Problem 7: Suppose (R, +, °) is a ring in which a ° a = 0 implies a = 0. Show that R has no nonzero nilpotent elements.Problem 8: Let A and B be ideals of a ring (R, +, °). If A n B = {0}, show that a ° b = 0 whenever a ? A and b ? B.Problem 9: Show that x4 + 1 is irreducible over Q but reducible over R. What is the smallest extension field containing Q for which x4+ 1 is reducible?Notation:NG(H) denotes the normalizer of H in G.CG(H) denotes the centralizer of H in G.Aut(H) denotes the class of all automorphisms of H.Z(G) denotes the center of G.U(165) denotes the group of units for 165.
	
Document Preview: Solve each problem:Problem 1: Let H be a subgroup of group (G, °). a.) Show that NG(H) / CG(H) is isomorphic to a subgroup of Aut(H). b.) Show that CG(H) = NG(H) if |H| = 2. Problem 2: Let G be a group of order |G| = 992 = 25 · 31. a.) Show that at least one of its Sylow p-subgroups is normal. b.) If G is Abelian, find all prime-power decompositions and invariant decompositions. Problem 3: Prove that if G / Z(G) is cyclic, then G is Abelian. Problem 4: Let G be a group of order 99. a.) Prove that there exists a subgroup of H of order 3. (Should be a one-sentence proof.) b.) Prove there is a unique subgroup K of G such that K / H ˜ Z3 and G / K ˜ Z11. Problem 5: Characterize those integers n such that the only Abelian groups of order n are cyclic. Problem 6: Express U(165) as an internal direct product of proper subgroups in 4 different ways. Problem 7: Suppose (R, +, °) is a ring in which a ° a = 0 implies a = 0. Show that R has no nonzero nilpotent elements. Problem 8: Let A and B be ideals of a ring (R, +, °). If A n B = {0}, show that a ° b = 0 whenever a ? A and b ? B. Problem 9: Show that x4 + 1 is irreducible over Q but reducible over R. What is the smallest extension field containing Q for which x4+ 1 is reducible? Notation: NG(H) denotes the normalizer of H in G. CG(H) denotes the centralizer of H in G. Aut(H) denotes the class of all automorphisms of H. Z(G) denotes the center of G. U(165) denotes the group of units for 165.

David · Accepted Answer

1. (a) We can define a map χ : NG(H)→ Aut(H) by sending g to the automorphism σg : H → H obtained
from conjugation by g. By definition of normalizer we have σg(h) = ghg
−1 ∈ H whenever h ∈ H and
g ∈ NG(H), so σg is well defined. Then g ∈ Ker(χ) if and only if σg(h) = h for all h ∈ H, if and only
if gh = hg for all h ∈ H, i.e, if and only if g ∈ CG(H). By the First Isomorphism Theorem we have
NG(H)/CG(H) ∼= Im(χ), and note that Im(χ) is a subgroup of Aut(H).
1. (b) Since |H| = 2, therefore H is abelian and hence all elements commute with each other and so CG(H) =
NG(H).
2. (a) The number of 31-Sylow subgroups must divide 31 × 32 and be congruent to 1 modulo 31; so either
there is a single 31-Sylow subgroup (in which case the group is not simple), or there are thirty two
31-Sylow subgroups.
If there are thirty two 31-Sylow subgroups, then since any two distinct ones must intersect trivially (the
groups are cyclic of prime order, so the only proper subgroup is trivial), they account for 32(31−1)+1
elements of G.
That means that there are 32 × 31 − 32 × 30 = 32 elements whose order is not 31.

Problem 1: Let H be a subgroup of group (G, °). a.) Show that NG(H) / CG(H) is isomorphic to a subgroup of Aut(H). b.) Show that CG(H) = NG(H) if |H| = 2. Problem 2: Let G be a group of order |G| =...

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