Math 360, Fall 2014, Assignment 3

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

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

1. Isomorphism (from one binary structure to another).
2. Isomorphic (binary structures).
3. Structural property (of a binary structure).
4. Identity element.
5. Inverse (of an element of some binary structure with an identity).
6. Group.
7. Abelian group.

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

1. Theorem concerning the uniqueness of identity elements (Theorem 3.13).
2. Theorem concerning the uniqueness of inverses in groups (second part of Theorem 4.17).

Solve the following problems:

1. Section 3, problems 2, 4, 8, 9, and 17.
2. Section 4, problems 1, 5, 7, 12, 13, and 17.

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Solutions:

In an effort to promote collaboration, I will only post work on problems whose problem numbers are elements of $[2]_{3}$. Please join me in working on the problems here. -Ian

Definitions:

3. A structural property of a binary structure $(S,\ast)$ is a property that holds for all structures that are isomorphic to $(S,\ast)$.
6. A group $(G,\ast)$ is a binary structure for which the following three properties hold:
   1. $\ast$ is associative
   2. $G$ contains an identity element $e$, i.e. for all $a\in G$, $a\ast e = e\ast a = a$
   3. For all $a\in G$, there exists $a'\in G$ with $a\ast a'=a'\ast a=e$

Theorems:

2. In any group $(G,\ast)$, for each $a\in G$, there is only one element $a'\in G$ with $a\ast a'=a'\ast a=e$, where $e$ is the identity element.