gallian - Solns to HW problems from Gallian Jonathan L.F...

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Solns to HW problems from Gallian Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 27 March, 2008 (at 15:34 ) Terms. Typical group notation: ( G, · , e ) or , · , ε ) or ( G, · , 1) or ( G, + , 0). The symbol for the neutral ( i.e, identity ) element may change, according to whether the group name is a Greek letter, or whether the group is written multiplicatively or additively. A vectorspace might be written as ( V , + , 0 ). A group of functions , under composition, might be written ( G, , Id ). We’ll use 11 ( a blackboard bold ‘1’ ) for the trivial group , but in specific cases may write { e } or { 0 } . Use Cyc N , S N , D N for the N th cyclic , symmetric and dihedral groups. So | Cyc N | = N and | S N | = N ! and | D N | =2 N . The alternating group A N has | A 1 | = 1; otherwise, | A N | is N ! / 2. Use Z( G ) for the cen- ter of G . The automorphisms of G form a group ( Aut( G ) , , Id ) . Each x G yields an inner automorphism of G defined by J x ( g ) := xgx 1 . The set { J x } x G is writ- ten Inn( G ); it is a normal subgp of Aut( G ). The map J : G Aut( G ) by J ( x ) := J x , is a group homor- phism. 1 : All Involution Lemma. If every element of ( G, · , e ) is an involution, then G is abelian. Pf. Fix a, b G . So e = [ ab ] 2 = aba 1 b 1 = a, b . For a ( possibly infinite ) group G and posint D , define S D,G := x G Ord( x ) = D . On S D,G define this relation: x D y IFF x G = y G . 2 : Phi Lemma. With S D,G and D from above: x D y IFF x y . In particular, each equiv- alence class has precisely ϕ ( D ) many elements. So ϕ ( D ) divides | S D,G | . Moreover, the ratio | S D,G | ϕ ( D ) equals the num- ber of cyclic order- D subgroups of G . Proof. By hypothesis, x y . But these sets have the same, finite , cardinality. So they are equal. An elt x G generates an order- D cyclic subgp IFF x S D,G . So the order- D cyclic subgroups are in 1-to-1 correspondence with the above equivalence classes. For y G we use Periods G ( y ) for the set of in- tegers k with y k = e . A subgroup H G de- termines a similar set. Let P H ( y ) = P H,G ( y ) be { k Z | y k H } . So Periods( y ) is simply P H ( y ), when H is the trivial subgp { e } . 3 : Periods Lemma. Fix G, H, y as above, and let P H mean P H ( y ) . If P H is not just { 0 } , then P H = N Z , where N is the least positive element of P H . Proof. Suppose N := Min( Z + P H ) is finite. Fixing a k P H , we will show that k |• N . Set D := Gcd( N, k ). LBolt ( well, ezout’s lemma ) produces integers such that D = NS + kT . Hence D P H , since y D equals [ y N ] S · [ y k ] T = e S · e T . Thus N = D •| k . 4 : Defn. Use H-Ord( y ) or H-Ord G ( y ) for the above N ; else, if P H is just { 0 } then H-Ord( y ) := . Call this the H -order of y . The order of y , written Ord( y ) or Ord G ( y ), is simply H-Ord G ( y ) when H := { e } . 5 : Lemma. Suppose that f :( G, e ) , ε ) is a group- homomorphism. Then TFAEquivalent: i : Ker( f ) is trivial , i.e, is { e } . ii : f is injective. iii : When Ord( G ) = Ord(Γ) < : f is surjective. # 37 P.
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