[]GNGʬȤ롣GG/Nؤμff(x)=xNˤäơСf:GG/NפϽƱǤ뤳Ȥ
[]fƱǤ뤳Ȥˤϡf(xy)=f(x)f(y)ΩĤȤǧФ褤
()
=f(xy)
=xyN
()
=f(x)f(y)
=(xN)(yN)
=xyN(G/NѤ)
äơ()=()ꡢfϽƱǤ롣
[]Ʊñ̸ñ̸˰ܤոո˰ܤ
[]G1,G2˷Ȥ롣
[1]Ⱦ
G1ñ̸e1G2ñ̸e2Ȥ롣
f:G1G2פƱǤȤȡf(e1)=e2ΩĤȤФ褤
G1ˤơe1e1=e1Ωġ
fǰܤȡf(e1)f(e1)=f(e1)Ȥʤ롣
f(e1)G2θǤ뤿ᡢG2ˤf(e1)εո餫ȡf(e1)=e2Ȥʤ롣
[2]Ⱦ
aG1ǤդθȤȤf(a-1)=(f(a))-1Ф褤
G1ˤơaa-1=e1Ωġ
fǰܤȡf(a)f(a-1)=f(e1)Ȥʤ롣
Ⱦη̤ꡢf(a)f(a-1)=e2
G2ˤơG2θf(a)εո餫ȡf(a-1)=(f(a))-1Ȥʤ롣
[]GaG1ĤθȤ롣
ˡZ鷲Gؤμff(x)=axСf:ZGϽƱˤʤ롣
[]ZǤդθx,yФơΩġ
f(x+y)
=ax+y
=(ax)(ay)
=f(x)f(y)
äơfϽƱǤ롣
[]f:G1G2G1鷲G2ؤƱȤ롣
G1θfˤäơG2ñ̸˰ܤΤνfγȤKer(f)ɽ
ޤG2θf(x) (xG1)ηɽΤνfȤf(G1)ޤIm(f)ɽ
ȤäɽȼΤ褦˽롣
Ker(f)={xG1|f(x)=e2}
[]f:G1G2G1鷲G2ؤνƱȤ롣
ΤȤfγKer(f)ϡG1ʬǤ롣
[]
[1]fγKer(f)G1ʬǤ뤳ȤΤ롣
äơKer(f)G1ʬˤʤ롣
[2]Ker(f)G1ʬǤ뤳ȤΤ롣
ʬꡢgG1,nKer(f) g-1ngKer(f)פΩĤȤǧФ褤
f(g-1ng)
=f(g-1)f(ng)ʢfϽƱǤ뤫
=f(g-1)f(n)f(g)ʢfϽƱǤ뤫
=f(g-1)e2f(g)ʢnKer(f)ꡢf(n)=e2
=f(g-1)f(g)ʢe2G2ñ̸Ǥ뤫
=f(g-1g)ʢfϽƱǤ뤫
=f(e1)ʢe1G1ñ̸Ǥ뤫
=e2ʢ[]ֽƱñ̸ñ̸˰ܤոո˰ܤפȾ
äơg-1ngKer(f)
ʾꡢKer(f)G1ʬǤ롣
[]f:G1G2G1鷲G2ؤνƱȤ롣
ΤȤff(G1)ϡG2ʬǤ롣
[]f(G1)G2θʤ뽸Ǥ뤿ᡢf(G1)G2ʬǤ뤿ˤϡ3Ф褤e2G2ñ̸Ȥ롣
[1]x,yf(G1)xyf(G1)
xy
=f(a)f(b)ʢx,yf(G1)Ȥȡx=f(a),y=f(b), a,bG1Ȥ˽뤫
=f(ab)ʢfϽƱǤ뤿
f(G1)
[2]e2f(G1)
e2
=f(e1)ʢ褳G1ñ̸e1Ȥơ[]ֽƱñ̸ñ̸˰ܤոո˰ܤפȾ
f(G1)
[3]xf(G1)x-1f(G1)
x-1
=(f(a))-1ʢx=f(a), aG1֤
=f(a-1)ʢ[]ֽƱñ̸ñ̸˰ܤոո˰ܤפθȾ
f(G1)