Flashcards
Ch. 4: Cyclic Groups
Cards in this deck (14)
- 1Def: Cyclic GroupA 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.
- 2Criterion for a^i = a^jLet 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
= {e,a,a^2,...,a^(n-1)}
and a^i = a^j iff n|(i-j) - 3Corollary 1: |a|=||For any group element a, |a|=||
- 4Corollary 2: a^k = e implies |a| divides ka^k = e implies |a| divides k
- 5Theorem: =Let a be an element of order n in a group and let k be a positive integer.
Then =
and |a^k| = n/(gcd(n,k) - 6Corollary 1: Oders of Elements in Finite Cyclic GroupsIn a finite cyclic group, the order of an element divides the order of the group
- 7Corollary 2: Criterion for = and for |a^i| = |a^j|Let |a| = n.
Then = iff gcd(n,i) = gcd(n,j)
And |a^i| = |a^j| iff gcd(n,i) = gcd(n,j) - 8Corollary 3: Generators of Finite Cyclic GroupsLet |a| = n.
Then = iff gcd(n,j) = 1
and |a| = |a^j| iff gcd(n,j) = 1 - 9Generators of Z_nAn integer in Z_n is a generator of Z_n iff gcd(n,k) = 1
- 10Fundamental Theorem of Cyclic GroupsEvery subgroup of a cyclic group is cyclic. Moreover, if || = n, then the order of any subgroup of is a divisor of n; and, for each positive divisor k of n, the group has exactly one subgroup of order k -- namely,
- 11Corollary Subgroups of Z_nFor each positive divisor k of n, the set if the unique subgroup of Z_n of order k; moreover, these are the only subgroups of Z_n.
- 12Def: Phi functionLet 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 - 13Number of elements of each order in a cyclic groupIf d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is Phi(d).
- 14Corollary: Number of elements of oder d in a finite groupIn a finite group, the number of elements of order d is divisible by Phi(d)
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- Mathematics (Abstract Algebra) Flashcards on Ch. 4: Cyclic Groups, created by William Hartemink on 09/04/2017.
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