Cyclic Groups: Generators, Orders, and Structural Properties

All flashcards

Cyclic group

A group whose elements are all integer powers of one element.

Order of an element

The number of elements in the cyclic subgroup generated by that element; it may be infinite.

Finite cyclic group of order $n$

A group isomorphic to the additive group $\mathbb{Z}/n\mathbb{Z}$.

Primitive root modulo $n$

A generator of the multiplicative group of units modulo $n$.

Subgroups of a finite cyclic group

Every subgroup is cyclic, and there is exactly one subgroup for each positive divisor of the group order.

Quiz questions

1. Why is every cyclic group abelian?Answer: Any two elements are powers of one generator, and their exponents commute under addition.
2. When does the residue class of $m$ generate the additive group $\mathbb{Z}/n\mathbb{Z}$?Answer: When $\gcd(m,n)=1$.
3. What is the order of the residue class of $m$ in $\mathbb{Z}/n\mathbb{Z}$?Answer: $\frac{n}{\gcd(n,m)}$
4. What is true of every quotient group of a cyclic group?Answer: It is cyclic.
5. For which condition are the finite cyclic groups' direct product $\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Z}/nm\mathbb{Z}$ isomorphic?Answer: When $n$ and $m$ are relatively prime.