8492820
Description
Flashcards by William Hartemink, updated more than 1 year ago
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Created by William Hartemink
about 7 years ago
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| Question | Answer |
| Def: Cyclic Group | A cyclic group is a group that can be generated by a single element. Cyclic groups are Abelian. A cyclic group of order n is often denoted Z_n, and its generator a satisfies a^n = e. |
| Criterion for a^i = a^j | Let a be an element of Group G. If a has infinite order, then a^i = a^j iff i=j. If a has finite order, say, n, then <a> = {e,a,a^2,...,a^(n-1)} and a^i = a^j iff n|(i-j) |
| Corollary 1: |a|=|<a>| | For any group element a, |a|=|<a>| |
| Corollary 2: a^k = e implies |a| divides k | a^k = e implies |a| divides k |
| Theorem: <a^k> = <a^(gcd(n,k)> | Let a be an element of order n in a group and let k be a positive integer. Then <a^k> = <a^(gcd(n,k)> and |a^k| = n/(gcd(n,k) |
| Corollary 1: Oders of Elements in Finite Cyclic Groups | In a finite cyclic group, the order of an element divides the order of the group |
| Corollary 2: Criterion for <a^i> = <a^j> and for |a^i| = |a^j| | Let |a| = n. Then <a^i> = <a^j> iff gcd(n,i) = gcd(n,j) And |a^i| = |a^j| iff gcd(n,i) = gcd(n,j) |
| Corollary 3: Generators of Finite Cyclic Groups | Let |a| = n. Then <a> = <a^j> iff gcd(n,j) = 1 and |a| = |a^j| iff gcd(n,j) = 1 |
| Generators of Z_n | An integer in Z_n is a generator of Z_n iff gcd(n,k) = 1 |
| Fundamental Theorem of Cyclic Groups | Every subgroup of a cyclic group is cyclic. Moreover, if |<a>| = n, then the order of any subgroup of <a> is a divisor of n; and, for each positive divisor k of n, the group <a> has exactly one subgroup of order k -- namely, <a^(n/k)> |
| Corollary Subgroups of Z_n | For each positive divisor k of n, the set <n/k> if the unique subgroup of Z_n of order k; moreover, these are the only subgroups of Z_n. |
| Def: Phi function | Let Phi(1) = 1 For n>1, Phi(n) denote the number of positive integers less than n and relatively prime to n. Phi(2)=1 Phi(3) = 2 Phi(4) = 2 Phi(5) = 4 Phi(6) = 2 Phi(7) = 6 |
| Number of elements of each order in a cyclic group | If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is Phi(d). |
| Corollary: Number of elements of oder d in a finite group | In a finite group, the number of elements of order d is divisible by Phi(d) |
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