## Character Table of S_3 Without Finding Irreducible Characters

**Point of post: **In this post we construct the character table of without having to actually find the irreducible characters.

*Motivation*

As was stated in our last post this post shall serve to show how nice the theory we’ve devoloped can make the construction of character tables. In particular, it is often very unapparent percisely how to construct all the irreducible characters. It is just an odd coincidence that for the irreps are so obvious. So, we shall show that in this post except for the trivial irrep which requires no thought to construct we don’t even need to construct either of the other two characters.

*Character Table of Without Finding the Irreducible Characters*

You’ll recall from our last post that we were able to ascertain that the character table for is of the form

Moreover, since and it clearly follows that one of the conjugacy classes of and contain three elements and the other two. But, it’s clear by inspection that and are conjugate to (not only by inspection, but a common theorem I will discuss later which says that two elements of a symmetric group are conjugate if and only if they have the same ‘cycle type’) and so and more importantly and so . But, from **CT 3 **we know that

Note next that by **CT 9**. Thus, by we may conclude that . But, by **CT 2 **we have (considering the first and third rows)

so that . Moreover, by considering **CT 2 **again between rows two and three we see that

and so . Finally, considering **CT 2 **one last time this time considering row two and row one we see that

and thus we may conclude that . Thus, we may finally conclude that the character table for is

*Consequences of the Character Table*

Using **CT 5 **and **CT 6** which are really just statements about the kernel and center of a character respectively we may conclude that the set of all normal subgroups of is equal to and the center is equal to .

**References:**

1. Isaacs, I. Martin. *Character Theory of Finite Groups*. New York: Academic, 1976. Print.

2. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

[…] groups and given the knowledge of the characters of and . So for example, it’s easy (as we’ve shown) to construct the character table for and it’s equally easy to construct the character table […]

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[…] begin by constructing the character tables for and . The first of these we have already done and is given […]

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