Math 360, Fall 2017, Assignment 10

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

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

1. Right congruence (modulo a subgroup $H$).
2. $Hg$ (i.e. the right coset of $H$ by g).
3. Normal subgroup.
4. $gHg^{-1}$ (i.e. the conjugate of $H$ by $g$).
5. Index (of a subgroup).
6. $G/H$ (a.k.a. the left coset space of $H$ in $G$).
7. $H\backslash G$ (a.k.a. the right coset space of $H$ in $G$).
8. Coset multiplication.

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

1. Theorem concerning subgroups of a group of prime order.
2. Theorem classifying groups of prime order.
3. Theorem describing the elements of $Hg$.
4. Theorem giving five alternative conditions equivalent to normality.
5. Theorem concerning normality of index two subgroups.
6. Theorem characterizing when coset multiplication is well-defined.

Solve the following problems:

1. Section 10, problems 12, 13, 15, and 34.
2. Section 14, problems 1, 9, 24, 31, and 34.
3. (Another helpful corollary of Lagrange's theorem) Suppose $G$ is a finite group, and $g\in G$. Prove that $g^{\left\lvert G\right\rvert}=e$. Give several concrete examples of this phenomenon.

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