Today I Learned

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

Monthly Archives: February 2017

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)|.