Today I Learned

Some of the things I've learned every day since Oct 10, 2016

108: Generating Set for the Symmetric Group

There are many possible choices of a generating set for $S_n$, the symmetric group. However, one particularly simple one is the set

$\{(0, 1, 2, \dots, n-1), (0, 1)\}$

where $0, 1, 2, \dots, n-1$ is an ordering of the elements being permutated by the elements of $S_n$, $(0, 1, 2, \dots, n-1)$ is the permutation ‘shifting’ all these elements to their successor, and $(0, 1)$ is the permutation swapping the first 2 of these.

I won’t provide a formal proof here as it’s a little tedious, but you can easily convince yourself that every permutation of these $n$ elements is a combination of these 2 permutations, because alternating between them in succession allows you to move any of the $n$ elements to an arbitrary position in the ordering.

107: The Order of the Product of 2 Group Elements

Where $G$ is a group, and $a, b \in G$, it is always the case that

$|ab| = |ba|$.

That is, we can speak of ‘the’ order of the product of 2 group elements without worrying about which product it is.

106: Group Isomorphisms

In group theory, a group isomorphism between 2 groups $G, H$ does not simply refer to an ‘isomorphism’ between the underlying sets of $G, H$, but to a special type of homomorphism between them: one which is bijective. In a sense, isomorphic groups are the same group, just with different symbols representing their elements and operations — something which is not necessarily true if they’re homomorphic or if just their underlying sets are isomorphic.

In particular, if $\varphi : G \mapsto H$ is a group isomorphism, then

• $|G| = |H|$
• $G$ is abelian $\leftrightarrow H$ is abelian
• for all $x \in G, |x| = |\varphi (x)|$.