既约剩余类群Z_M^*+)的循环子群

既约剩余类群Z M * + )的循环子群 CYCLIC SUBGROUPS OF THE PRIME RESIDUE GROUP Z M * 方华鹏 Fang Huapeng first-author < 正 > §1 引言在[1]中,F.Rudolf Begl曾得到如下两个结论: 1.设p是一个奇素数,m≥1,M=p m ,则（ⅰ）:既约剩余类r在Z M * 的西洛p-子群内的充要条件是p|（r-1）,若令r的阶为p q ,则又有q=max（m-#p（r-1）,0）,其中#p（n）表示能整除n的p的最高次幂的指数。（ⅱ）:Z M * 的西洛p-子群P是循环群,其生成元是1+vp,（v,p）=1。（ⅲ）:设r、s为Z M * 中阶为p幂的元,即r、s∈P,则下列三个命题等价: ①r和s生成相同的循环子群; In this paper we prove the following two principal theorems: Theorem 5 Let M=2 k p 1 α 1 …p t α t be the prime factorization, Z M * be the prime residue group modulo M, P j be a Sylow p j -subgroup of Z M * , k≥0, t≥1, and suppose （p j , p i -1）=1 for any i, j, then （ⅱ） Pj={α} is a cyclic subgroup of order p j ( α j -1 of Z M * , in which any generating element must be of the form α=1+lp j , where（l, p j ）=1, 2 k P i α 1 …P j-1 α j-1 p j+1 α j+1 …p t α t |l; （ⅲ）If r、s∈P j , then the following three conditions are equivalent: ① r and s generate the same cyclic subgroup; ② r and s have the same order in Z M * ; ③ (r-l, p j α j )=(s-1, p j α j ); （ⅳ） when r∈P j , 1≤r≤M, then o（r）=p j α j-u , where μ satisfies the condition (r-1, p < sup > α j )=p j μ . Remark In the theorem 5, if k=0, i=1, then the result reduces to that, obtained in [1] of F. Rudolf Begl. Theorem 6 If α≥4, then Z 2 α * can be represented as a direct product of cyclic subgroups of Z 2 α * in four dictinct ways, such as or Z 2 α * =U 2 ×{2 α-1 -1}, where U 1 ={5+8k}, U2={3+8h} are cyclic subgroups of Z 2 α * and k is an arbitrary integer. To prove the theorem 6, we need the following lemma, Lemma 3 Let a be an element of order 2 α-2 of Z 2 α * （a≥3）, we have （ⅰ） when α=3, then α≡3, 5, 7 （mod 8）; （ⅱ） when α≥4, then α≡3, 5 （mod 8）. Remark In theorem 6 and lemma 3, we have extended the results of [1] and [3] . 1981-04-01 2021-04-01 4