S4S5的子群文档格式.docx
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S4S5的子群文档格式.docx
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(24)>
={
(1),(24)},
N6=<
(14)>
={
(1),(14)},
N7=<
(34)>
={
(1),(34)},
N8=<
(12)(34)>
={
(1),(12)(34)},
N9=<
(13)(24)>
={
(1),(13)(24)},
N10=<
(14)(23)>
={
(1),(14)(23)},
其中N2至N7为一共轭类,N8至N10为一共轭类。
3阶子群:
由S4的3阶元生成的循环群,因为每两个互逆的3阶元同单位元可以组成一个子群,而S4的3阶元有6个,所以S4的3阶子群有3个,且为一共轭类,即:
N11=<
(123)>
={
(1),(123),(132)},
N12=<
(134)>
={
(1),(134),(143)},
N13=<
(124)>
={
(1),(124),(142)},
N14=<
(234)>
={
(1),(234),(243)},
4阶子群:
(循环群和非循环群)
循环群:
由S4的4阶元生成的循环群,根据生成的子群的元的情况,一个4阶元生成的子群里包含有一对互逆的4阶元,而S4的4阶元有三对互逆的元,故4阶循环子群有3个,且为一共轭类,即:
N15=<
(1234)>
={
(1),(1234),(13)(24),(1432)},
N16=<
(1324)>
={
(1),(1324),(12)(34),(1423)},
N17=<
(1243)>
={
(1),(1243),(14)(23),(1342)},
非循环群:
其元都为2阶元,且两个互不相同的2阶元相乘可得另一个2阶元,满足这一条件可构成的4阶非循环群只有4个,且为2个共轭类,即:
N18={
(1),(12),(34),(12)(34)},
N19={
(1),(13),(24),(13)(24)},
N20={
(1),(14),(23),(14)(23)},
和N21={
(1),(12)(34),(13)(24),(14)(23)}
8阶子群:
此群里的元的阶只能为1阶、2阶、4阶,且由sylow定理,8阶子群里必含有4阶子群,故可先确定8阶子群里的4个元素,其余4个元素可由已确定的元来给出,经由此算法,由全部的4阶子群只找出3个8阶子群,故8阶子群有3个,且为一共轭类,即:
N22={
(1),(1234),(13)(24),(1432),(13),(12)(34),(24),
(14)(23)},
N23={
(1),(1324),(12)(34),(1423),(12),(13)(24),(34),
(14)(32)},
N24={
(1),(1243),(14)(23),(1342),(14),(12)(43),(23),
(13)(24)},
24阶子群:
即N25=S4
以上为S4里必存在的子群,下面讨论S4里可能存在的子群:
6阶子群:
因为S4包含着S3,故S4必有同构于S3的一类6阶子群,而同构于S3的S4的6阶子群有4个,且其元为1阶、2阶和3阶,所以S4的6阶子群有4个,且为一共轭类,即:
N26={
(1),(12),(13),(23),(123),(132)},
N27={
(1),(12),(24),(14),(124),(142)},
N28={
(1),(34),(13),(14),(143),(134)},
N29={
(1),(34),(24),(23),(234),(243)},
12阶子群:
若S4有12阶子群,则由sylow定理,该子群里必存在2阶子群、4阶子群和3阶子群,经计算,S4的12阶子群只有一个,即:
N30={
(1),(123),(132),(134),(143),(124),(142),(234),(243),(12)(34),(13)(24),(14)(23)}。
综上,S4共有30个子群,分为10个共轭类,其中,由正规子群定义及定理6知S4的1阶子群,N21,12阶子群和24阶子群为正规子群。
§
3.2S5的元
已知|S5|=120及S5的的元的形式为(a),(ab),(abc),(abcd),(abcde),(ab)(cd),(ab)(cde)其中a,b,c,d,e∈{1,2,3,4,5}
因为(a)=(b)=(c)=(d)=(e),所以1阶元有1个,即单位元
(1);
形式为(ab)或(ab)(cd),共有C52+
(C52C32)=25个,即:
(12),(13),(14),(15),(23),(24),(25),(34),(35),(45),(12)(34),(12)(35),(12)(45),(13)(24),(13)(25),(13)(45)(14)(23),(14)(25)(14)(35)(15)(23),(15)(24),(15)(34),(23)(45),(24)(35),(25)(34);
3阶元:
形式为(abc),共有C53A22=20个,即:
(123),(124),(125),(132),(134),(135),(142),(143),(145),(152),
(153),(154),(234),(235),(243),(245),(253),(254),(345),(354);
4阶元:
形式为(abcd),共有C54A33=30,即:
(1234),(1235),(1243),(1245),(1253),(1254),(1324),(1325),(1342),
(1345),(1352),(1354),(1423),(1425),(1432),(1435),(1452),(1453),
(1523),(1524),(1532),(1534),(1542),(1543),(2345),(2354),(2435),
(2453),(2534),(2543);
5阶元:
形式为(abcde),共有C55A44=24,即:
(12345),(12354),(12435),(12453),(12534),(12543),(13245),(13254),(13425),(13452),(13524),(13542),(14235),(14253),(14325),(14352),(14523),(14532),(15234),(15243),(15324),(15342),(15423),(15432);
6阶元:
形式为(ab)(cde),共有C52C33A22=20,即:
(12)(345),(12)(354),(13)(245),(13)(254),(14)(235),(14)(253),
(15)(234),(15)(243),(23)(145),(23)(154),(24)(135),(24)(153),(25)(134),(25)(143),(34)(125),(34)(152),(35)(124),(35)(142),
(45)(123),(45)(132);
3.3S5的子群
因为|S5|=120,由定理1,知S5子群的阶可能为:
1,2,3,4,5,6,8,10,12,15,20,
24,30,40,60,120,又因为|120|=23×
3×
5,根据sylow定理,S5必存在2阶、3阶、4阶、5阶和8阶子群,另S5有平凡子群1阶子群和120阶子群,可能有6阶、10阶、12阶、15阶、20阶、24阶、30阶、40阶和60阶子群。
下述S5的各个阶子群的情况:
S5的一阶子群为平凡子群,只包含单位元
(1),即
H1={
(1)}。
由S5的2阶元生成的循环群,由于2阶子群里只有两个元,其中一个为单位元,由定理2,可知另一个元必为2阶元,因为S5共有25个二阶元,所以S5共有25个2阶子群,其中分为两个共轭类,第一个共轭类为:
H1=<
H2=<
H3=<
={
(1),(14)},
H4=<
(15)>
={
(1),(15)},
H5=<
={
(1),(23)},
H6=<
={
(1),(24)},
H7=<
(25)>
={
(1),(25)},
H8=<
={
(1),(34)},
H9=<
(35)>
={
(1),(35)I,
H10=<
(45)>
={
(1),(45)};
第二个共轭类为:
H11=<
H12=<
(12)(35)>
={
(1),(12)(35)},
H13=<
(12)(45)>
={
(1),(12)(45)},
H14=<
H15=<
(13)(25)>
={
(1),(13)(25)},
H16=<
(13)(45)>
={
(1),(13)(45)},
H17=<
H18=<
(14)(25)>
={
(1),(14)(25)},
H19=<
(14)(35)>
={
(1),(14)(35)},
H20=<
(15)(23)>
={
(1),(15)(23)},
H21=<
(15)(24)>
={
(1),(15)(24)},
H22=<
(15)(34)>
={
(1),(15)(34)},
H23=<
(23)(45)>
={
(1),(23)(45)},
H24=<
(24)(35)>
={
(1),(24)(35)},
H25=<
(25)(34)>
={
(1),(25)(34)}。
3阶子群:
由S4的3阶元生成的循环群,由定理1,3阶子群里元的阶只可能为1阶和3阶,S5里1阶的元为单位元,3阶元有20个,3阶元恰好有10组互为逆元,经计算S5的3阶子群有10个,为一个共轭类,即:
={
(1),(123),(132)},
H2=<
={
(1),(124),(142)},
H3=<
(125)>
={
(1),(125),(152)},
H4=<
={
(1),(134),(143)},
H5=<
(135)>
={
(1),(135),(153)},
H6=<
(145)>
={
(1),(145),(154)},
H7=<
={
(1),(234),(243)},
H8=<
(235)>
={
(1),(235),(253)},
H9=<
(245)>
={
(1),(245),(254)},
H10=<
(345)>
={
(1),(345),(354)}。
4阶子群:
由S5的4阶元生成的循环群,根据生成的子群的元的情况,一个4阶元生成的子群里包含有一对互逆的4阶元,而S5的4阶元有三对互逆的元,故4阶循环子群有3个,且为一共轭类,即:
H1=<
(1235)>
={
(1),(1235),(13)(25),(1532)},
(1253)>
={
(1),(1253),(15)(23),(1352)},
(1245)>
={
(1),(1245),(14)(25),(1542)},
(1254)>
={
(1),(1254),(15)(24),(1452)},
(1325)>
={
(1),(1325),(12)(35),(1523)},
(1345)>
={
(1),(1345),(14)(35),(1543)},
(1354)>
={
(1),(1354),(15)(34),(1453)},
H11=<
(1435)>
={
(1),(1435),(13)(45),(1534)},
(1524)>
={
(1),(1524),(12)(45),(1425)},
(2345)>
={
(1),(2345),(24)(35),(2543)},
(2354)>
={
(1),(2354),(25)(34),(2453)},
(2435)>
={
(1),(2435),(23)(45),(2534)}。
其元都为2阶元,且两个互不相同的2阶元相乘可得另一个2阶元,满足这一条件可构成的4阶非循环群有20个,且为2个共轭类,即:
H16={
(1),(12),(34),(12)(34)},
H17={
(1),(12),(35),(12)(35)},
H18={
(1),(12),(45),(12)(45)},
H19={
(1),(13),(24),(13)(24)},
H20={
(1),(13),(25),(13)(25)},
H21={
(1),(13),(45),(13)(45)},
H22={
(1),(14),(23),(14)(23)},
H23={
(1),(14),(25),(14)(25)},
H24={
(1),(14),(35),(14)(35)},
H25={
(1),(15),(23),(15)(23)},
H26={
(1),(15),(24),(15)(24)},
H27={
(1),(15),(24),(15)(24)},
H28={
(1),(23),(45),(23)(45)},
H29={
(1),(24),(35),(24)(35)},
H30={
(1),(25),(34),(25)(34)},
H31={
(1),(12)(34),(13)(24),(14)(23)},
H32={
(1),(12)(35),(13)(25),(15)(23)},
H33={
(1),(12)(45),(14)(25),(15)(24)},
H34={
(1),(13)(45),(14)(35),(15)(34)},
H35={
(1),(23)(45),(24)(35),(25)(34)};
其中H16至H30为一共轭类,H31至H35为另一共轭类。
5阶子群:
由定理1,5阶子群里元的阶只能为1阶和5阶,故S5的5阶子群为由S5的5阶元生成的循坏群,S5里1阶的元为单位元,5阶元有24个,经计算S5的5阶子群有6个,为一个共轭类,即:
(12345)>
={
(1),(12345),(13524),(14253),(15432)},
(12354)>
={
(1),(12354),(13425),(15243),(14532)},
(12435)>
={
(1),(12435),(14523),(13254),(15342)},
(12453)>
={
(1),(12453),(14325),(15234),(13542)},
(12534)>
={
(1),(12534),(15423),(13245),(14352)},
(12543)>
={
(1),(12543),(15324),(14235),(13452)}。
S5的8阶子群:
由定理1,8阶子群里元的阶只能为1阶、2阶和4阶,仿照S4的8阶子群,经计算,S5有15个8阶子群,一个共轭类,即:
H1={
(1),(1234),(13)(24),(1432),(13),(12)(34),(24),(14)(23)},
H1={
(1),(1324),(12)(34),(1423),(12),(13)(24),(34),(14)(32)},
H1={
(1),(1243),(14)(23),(1342),(14),(12)(43),(23),(13)(24)},
H1={
(1),(1235),(13)(25),(1532),(13),(12)(35),(25),(15)(23)},
H1={
(1),(1245),(14)(25),(1542),(14),(12)(45),(25),(15)(24)},
H1={
(1),(1253),(15)(23),(1352),(15),(12)(53),(23),(13)(25)},
H1={
(1),(1254),(15)(24),(1452),(15),(12)(54),(24),(14)(25)},
H1={
(1),(1325),(12)(35),(1523),(12),(13)(25),(35),(15)(32)},
H1={
(1),(1345),(14)(35),(1543),(14),(13)(45),(35),(15)(34)},
H1={
(1),(1354),(15)(34),(1453),(15),(13)(54),(34),(14)(35)},
H1={
(1),(1425),(12)(45),(1524),(12),(14)(25),(45),(15)(42)},
H1={
(1),(1435),(13)(45),(1534),(13),(14)(35),(45),(15)(43)},
H1={
(1),(2345),(24)(35),(1543),(24),(23)(45),(35),(25)(34)},
H1={
(1),(2354),(25)(34),(2453),(25),(23)(45),(34),(24)(35)},
H1={
(1),(2435),(23)(45),(2534),(23),(24)(35),(45),(25)(34)}。
S5的120阶子群:
S5的120阶子群即为S5本身。
以上为S5里必存在的子群,下面讨论S5里可能存在的子群:
S5的6阶子群:
由定理1,6阶子群里元的阶只能为1阶、2阶和3阶,同构于S3的为一个共轭类,即:
H1={
(1),(12),(13),(23),(123),(132)},
H2={
(1),(12),(14),(24),(124),(142)},
H3={
(1),(12),(15),(25),(125),(152)},
H4={
(1),(13),(15),(35),(135),(153)},
H5={
(1),(13),(14),(34),(143),(134)},
H6={
(1),(14),(15),(45),(145),(154)},
H7={
(1),(23),(25),(35),(235),(253)},
H8={
(1),(24),(25),(45),(245),(254)},
H9={
(1),(24),(23),(34),(234),(243)},
H10={
(1),(34),(35),(45),(345),(354)},
由2阶元,3阶元,和2×
3循环置换构成一个共轭类,即:
H11={
(1),(12)(345),(12)(354),(12),(345),(354)},
H12={
(1),(13)(245),(13)(254),(13),(245),(254)},
H13={
(1),(14)(235),(14)(253),(14),(235),(253)},
H14={
(1),(15)(234),(15)(243),(15),(234),(243)},
H15={
(1),(23)(145),(23)(154),(23),(145),(154)},
H16={
(1),(24)(135),(24)(153),(24),(135),(153)},
H17={
(1),(25)(134),(25)(143),(25),(134),(143)},
H18={
(1),(34)(125),(34)(152),(34),(125),(152)},
H19={
(1),(35)(124),(35)(142),(35),(124),(142)},
H20={
(1),(45)(123),(45)(132),(45),(123),(132)}。
由3阶元,和2×
2循环置换构成一个共轭类,即:
H21={
(1),(12)(34),(345),(354),(12)(35),(12)(45)},
H22={
(1),(14)(23),(235),(253),(14)(25),(14)(35)},
H23={
(1),(15)(23),(234),(243),(15)(24),(15)(34)},
H24={
(1),(13)(24),(245),(254),(13)(25),(13)(45)},
H25={
(1),(12)(34),(152),(125),(15)(34),(25)(34)},
H26={
(1),(12)(45),(123),(132),(13)(45),(23)(45)},
H27={
(1),(12)(35),(124),(142),(14)(35),(24)(35)},
H28={
(1),(13)(25),(134),(143),(14)(25),(25)(34)},
H29={
(1),(13)(24),(135),(153),(15)(24),(24)(35)},
H30={
(1),(14)(23),(145),(154),(15)(23),(23)(45)}。
所以S5的6阶子群共有30个。
S5的10阶子群:
若S5的10阶子群存在,则由Sylow定理,该10阶子群里必存在5阶子群,且此5阶子群亦为S5的5阶子群,又因为S5的10阶子群里的元的阶只能为1阶、2阶和5阶,所以经计算S5的10阶子群有6个,为一共轭类,即:
H1={
(1),(12345),(13524),(14253),(15432),(12)(35),(13)(45),
(14)(23),(15)(24),(25)(34)},
H2={
(1),(12354),(13425),(15243),(14532),(12)(34),(13)
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