Groups Lecture 8

in which we show that the sign of a permutation is a surjective group homomorphism, and meet cosets.

Having finished last time the proof that “sign is well-defined”, we could today in fact define the homomorphism $\mathrm{sgn}\colon S_n\to \{1,-1\}$ by $\mathrm{sgn}(\sigma)=(-1)^l$, where $l$ is the number of transpositions we need to get $\sigma$. All the hard work is in showing that it is really well defined! After that it is easy to show it is indeed a surjective group homomorphism. We call the kernel of $\mathrm{sgn}$ the alternating group $A_n=\mathrm{Ker}(\mathrm{sgn})$. As last result of the second chapter, we reminded ourselves (or perhaps “preminded”) that any subgroup of $S_n$ contains either no odd permutations or exactly half its permutations are odd. That is Question 3 on Sheet 2.

We then started on Chapter 3 and defined cosets: For a subgroup $H\leq G$ and an element $a\in G$, the left coset is $aH:= \{ah\mid h\in H\}$, and the right coset is $Ha:= \{ha\mid h\in H\}$. We saw some examples in $\mathbb{Z}$ and in $S_3\cong D_6$. To prepare for Lagrange’s Theorem, we defined a partition of a set $X$ to be a collection of subsets $X_i$ such that “the union gives all of $X$” (or “every element is in at least one $X_i$”) and the $X_i$ are pairwise disjoint (or “every element is in at most one $X_i$”).  The most important result of this chapter is Lagrange’s Theorem: For a finite group $G$, the size of any subgroup $H\leq G$ divides the size of $G$. We prove this by showing that the left cosets of $H$ partition $G$, and all have the same size (as $H$). We will finish that proof next time.

Understanding today’s lecture

It is probably good practice if you get used to seeing fairly quickly what the sign of a certain permutation is. Start by some single cycles. When you move to more general permutations in disjoint cycle notation, remember that sign is a homomorphism! So “even times even is even” because “plus times plus is plus”, and “odd times even is odd” because “minus times plus is minus” and so on.

To get used to cosets, there are a few things you could do. You have plenty of examples of subgroups: write down the cosets for them. Check if you can find a pattern when the right and left cosets might be the same or different. Remember that cosets are really sets and not subgroups (except the subgroup itself viewed as a coset).

Preparing for Lecture 9

See if you can finish the proof of Lagrange yourself. Can you think of ways Lagrange might be useful? For examples, does it tell you anything about the orders of elements?

Going a little deeper

The word “well-defined” is always a little tricky the first few times you meet it. What exactly does it mean? In fact, it can mean several things in different contexts. Here it means the following: our definition of the sign homomorphism seems to depend on a choice, namely the choice of how we write $\sigma$ as a product of transpositions. So if I made a different choice, would I get a different sign? That would not be a very good function! (In fact, it would not be a function.) So we have to check that the outcome, namely the $\mathrm{sgn}(\sigma)=(-1)^l$, does not depend on this choice, but that we get the same outcome whichever choice we made. This is what “well-defined” means here: it really is a function. We will meet “well-defined”ness again later, and it can mean in different contexts: does the function really land where we say it lands? Or: if we define something using a representative, do we get the same answer regardless of which representative we use (that is very close to the sense we used it today)? Look out for it in “Applications of Lagrange”, which comes in Chapter 3.