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

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern