MTH3003 Lecture 4

Simon Smith

Lucky for you, humans can’t do very much of this.

Following on from cyclic groups, we can generate a group $G$ with some generator $g$; this is then denoted $⟨g⟩$ (the smallest subgroup of $G$ that contains $g$). However, this isn’t restricted to a single generator element!

Groups Generated by Sets

If we extend the definition of a cyclic group to include more generators, e.g., $⟨g_1,g_2⟩$, we would create a group the exact same way. Formally, we define these groups generated by sets by saying…

Let $G$ be a group and let $S\subseteq G$. Then the group generated by $S$ is $⟨S⟩=\{a_1^{n_1}{a}_2^{n_2}\cdots a_m^{n_m}:m\in \mathbb{N}\text{ and }{n}_i\in \mathbb{Z}\text{ and }{a}_i\in S\text{ for all }1\le i\le m\}$. Or, similarly to before, the smallest subgroup of $G$ that contains all elements in $S$.

Klein four-group

The smallest non-cyclic group, denoted by $K_4$, is defined to be $K_4=⟨(12),(34)⟩$ - a set generated by two 2-cycle permutations.

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Pre-Lecture Notes from University Notes

• We can expand the definition of a cyclic group to have more than one generator; for example, $⟨g_1,g_2⟩$ - but this can be extended indefinitely. Basically it.