{ 萬年考古題 ＸＤ }
定義部份+解答
1.semi-group(半群):
{G,0} is called a semi-group.
if for all a,b belong to G => (a。b)。c = a。(b。c)
2.group(群)
{G,0} is called a group.
<1> for all a.b belong to G => a。b also belong to G.
<2> for all a.b.cbelong to G => (a。b)。c = a。(b。c)
<3> exist e belong to G,and for all a belong to G => e。a = e = a。e
<4> for all a belonf to G, and exist a。a^(-1) = a^(-1)。a = e
3.abelian(commutative) group(交換群)
a group G is called ab abelian group , if
for all a.b belong to G => a。b = b。a
4.order(秩)
G be a group is called the order of G is the number of
element of element and denoted by 。(G) . [or |G|]
5.symmetric group (對稱群)
The symmtric group of degree n is the set of all 1-1 , mappings
of a set n element onto itself and denoted by Sn , then clearly:
。(Sn) = n！
6.subgroup(子群)
G is a group H≦G , then H is called subgroup ,if H itself is
(包含於)
a group & denoted by H < G.
7.cyclic group(循環群)
A group G is called cyclic group generated by g ,if
exists g belong to G , such that G={g^n|n belong to Z} and denoted
by G=<g>.
8.coset(伴集)
If H<G , a belong to G ,then:
∕Ha = {ha|h belong to G} is called a right coset of H by a.
﹨aH = {ah|h brlong to G} is called a left coset of H by a.
9.index(指標)
If H<G ,the index of H in G is the number of distinct right(left)
cosets of H in G and denoted by indG(H)
(ie. G is a finite group ,then indG(H)= 。(G) )
───
。(h)
10.Largrage Theorem
If G is a finite froup and H is a subgroup of G ,
then 。(H)|。(G). (。(H) is a divisor of 。(G) ).
11.normal subgroup(正規子群)
H<G ,H is said to be a normal subgroup of G,
if for all g belong to H, g belong to G => g^(-1)hg belong to H
and denoted by H△G (△一個角是朝左,好奇怪,打不出來,符號一直變問號)
12.Quotient group(商群)
H△G (同上一題,一個角朝左,像小於的形狀), a set G/H = {Hg|g belong to G}
is called a quotient of G by H .
13.center
G is a group. Z(G)={ Z belong to G| ZX=XZ, for all X belong to G }
is called the center(or zenter) of G.
14.normalizer
H<G, N(H) = {Hg=gH| g belong to G} is called the normalizer of a in G.
15.centralizer
G is a group, a belong to G, Ca = {x|ax=xa , x belong to a} is called
the centralizer of a.
16.homomorphism(同態函數)
{G,0},{G,*} are two groups, if
for all x,y belong to G => f(x。y) = f(x)*f(y) , then f is called a
homomorphism from G into H.
17.isomorphism(同構函數)
A homomorphism f is called isomorphism if there f is one-to-one .
18.isomorphic(同構)
{G,0},{H,*} are two groups ,G.H are called isomorphic if there is an
isomorphism from G into H .
19.permutation(重排)
A bijection(1-1&onto) from a set S into itself is called a permution of S.
20.Cayley's Theorem
Every group is isomorphic to a permutation froup.
21.simple
A group is said to be simple if it has no homomorphic image.
(ie.if it has no nontrivial normal subgroups)
22.First Isomorphism Theorem
G.H two groups. δ: G→H (homomorphism) with Kerδ=K, then G/K=H .
23.alternating group(變換群)
An is called the alternating group of degree n, then An is the set of
all even permutation of Sn .
24.Conjugate(共軛)
G is a group a.b belong to G, then a.b is called conjugates if there
exist c belong to G such that b = c^(-1)ac ,and denoted by a～b .
25.class equation
If G is a finite group, then Ca = 。(G)/。(N(a)) ,
(ie.the number of lelments conjugate to a in G is the index of the
normalizer of a in G)
。(G)
=> 。(G) = Σ ────
。(N(a))
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◆ From: 61.228.145.232
※ 編輯: shihweng 來自: 61.228.151.237 (05/09 02:05)