#1 Cyclic Groups

One of the first concepts I came across in Linear Algebra were Groups. Groups are roughly defined as a set of values $G$, for which a function $\cdot: G \times G \to G$ is defined to combine the values in the set, in order to receive another value from the set (think of multiplication of numbers if you don't know about groups). The function has to be Associative, which means $(g \cdot h) \cdot k = g \cdot (h \cdot k)$, $\forall \{ g, h, k \} \subseteq G$. There exists one (and only one) so called Neutral Element $e \in G$, that leaves every other element $g \in G$ in function $\cdot$ unchanged: $e \cdot g = g, \quad \forall g \in G$. Beyond that, there must exist so called Inverse Elements $g^{-1} \in G$ for each element $g \in G$. For those elements applies $g \cdot g^{-1} = e$. I don't want to write too much about Groups and would therefore leave it by this imprecise explanation. If you'd like something more tangible you can have a look at the respective Wikipedia article.

Now what are Cylic Groups? Cyclic Groups are Groups that can generate itself from one element $g \in G$. What does generate mean? All elements can be reached by combining $g$ with itself (potentially) multiple times which would lead to something like this: $g^{3} \in G$ with $g^{3} = g \cdot g \cdot g$. This obviously has the effect, that from one element, every other element in the Group can be reached. Not all Groups are Cyclic Groups, but all Groups whos quantity of Elements is a prime number are Cyclic Groups. One example where Cyclic Groups show up is addition under modulo.