Question

• You should be able to prove the following statements: - If y : G → J be a group homomorphism andrEG has finite order, then ord(r) is divisible by ord(y(a)). If p: G → J is a group isomorphism and r e G, then r has infinite order if and only if ø(r) has infinite order. If G, J and G2 J2, then G1 x G2 J x J. If G is a cyclic group and K is subgroup of G, then K 4G and the quotient group G/K is also cyclic.

Answer 1

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E -braib) plaid) - dio homomorphism. I eu ,e are Identifier Ji fji Kero $ = {laib) EGIX G2 / carb) = reiieren? ={10.b) EG X G7] (0,(a), 42(b)) = leliei)] ={1916) 66X GR $,cal se'; $2162163 = {(a,b) & GX G2 / a=e, of b=e]. 1:0, fb2 are ={10,61). isomorphism) - dio one-one. Let cord) & J, XJ - CoJi & dfJ2 & 7 arG f beta (9)-f $2CD):d 1°! Diffs are isomorphism ) of Olaib) = (0, (a), 2(b)) - 161d) .ܗ܀ ܗܘ ܠܪ 8 ܙ 6 Cyclic - A G is cyclic group fo kio Subgroup of a - kit aloge subgroup. of G A RAG if G=Lay a k= camy for Some in of the = {gok 196 G} 1920 [G!K]=m. let (8K)m gmx as gfG I g=al 1. = gem k =k ( gemer) a order (96) m is finite, of less than or equal

of coclor m . A is a fute group of order m. f (ak) "= anak,-k Dank Ek back miln .n=rh. 867. & order of akymi & ord(ak)=m a G = Laky as is Cyclic.