if is this cyclic group definition? please explain
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Step-by-step explanation:
A cyclic group is a group that can be generated by a single element X (the group generator). Cyclic groups are Abelian.
A cyclic group of finite group order n is denoted C_n, Z_n, Z_n, or C_n; Shanks 1993, p. 75), and its generator X satisfies
X^n=I,
(1)
where I is the identity element.
The ring of integers Z form an infinite cyclic group under addition, and the integers 0, 1, 2, ..., n-1 (Z_n) form a cyclic group of order n under addition (mod n). In both cases, 0 is the identity element.
There exists a unique cyclic group of every order n>=2, so cyclic groups of the same order are always isomorphic (Scott 1987, p. 34; Shanks 1993, p. 74). Furthermore, subgroups of cyclic groups are cyclic, and all groups of prime group order are cyclic. In fact, the only simple Abelian groups are the cyclic groups of order n=1 or n a prime (Scott 1987, p. 35).
The nth cyclic group is represented in the Wolfram Language as CyclicGroup[n].
Examples of cyclic groups include C_2, C_3, C_4, ..., and the modulo multiplication groups M_m such that m=2, 4, p^n, or 2p^n, for p an odd prime and n>=1 (Shanks 1993, p. 92).
CyclicGroupTable
Cyclic groups all have the same multiplication table structure. The table for C_(20) is illustrated above.
By computing the characteristic factors, any Abelian group can be expressed as a group direct product of cyclic subgroups, for example, finite group C2×C4 or finite group C2×C2×C2. It is common to combine the indices for the highest prime factors of the direct product representation of a group since this provides a shorter notation and no ambiguity arises. For example C_2×C_3 is commonly written C_6.
The cycle index of the cyclic group C_p is given by
Z(C_p)=1/psum_(k|p)phi(k)a_k^(p/k),
(2)
where k|p means k divides p and phi(k) is the totient function (Harary 1994, p. 184). The first few are given by
Z(C_1) = x_1
(3)
Z(C_2) = 1/2x_1^2+1/2x_2
(4)
Z(C_3) = 1/3x_1^3+2/3x_3
(5)
Z(C_4) = 1/4x_1^4+1/4x_2^2+1/2x_4
(6)
Z(C_5) = 1/5x_1^5+4/5x_5.