Math 360, Fall 2013, Assignment 7

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

Carefully define the following terms, then give one example and one non-example of each:

1. Left congruence (i.e. the relation $\sim_L$ discussed on page 97 of the text).
2. Right congruence.
3. Left coset.
4. Right coset.
5. Index of a subgroup.

Carefully state the following theorems (you need not prove them):

1. Lagrange's Theorem.
2. Theorem concerning the indices of nested subgroups (Theorem 10.14 in the text).

Solve the following problems:

1. Section 10, problems 1, 7, 13, 15, 22, and 24.

--------------------End of assignment--------------------

Solutions:

Definitions

1.) Left Congruence ($\sim_{L,H}$)
Fix any group G, with a subgroup H. Define the relation, denoted $\sim_{L,H}$:
$a\sim_{L,H}b \Leftrightarrow b^-1a\in H$
We call this relation left congruence modulo H. Note that this is always an equivalence relation.(Proof)
Example: Let $G=S_3, H=A_3$. $(1,2,3) \sim_{L,H} (23)(21)$, since $(21)(23)(123)=(132)\in H$
Non-example: Let G and H be defined as above. $(1)(2)(3)\ \neg \sim_{L,H} (13)$, since $(13)(1)(2)(3)=(13) \notin H$.