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Introduction To Groups

Symmetries of an equilateral triangle, a braided circular pattern and a finite point set, motivating closure, identity and inverses before groups are defined formally.

A Complete Introduction to Abstract Algebra, Introduction To Groups

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3 exercises

Exercise 1

How many symmetries are there in the symmetric group of degree 6? In other words, what is the order of the symmetric group of degree 6?(Hint: Consider the number of permutations in the group.)

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Problem 1

Determine whether the following structures are groups. If they are, prove it. If not, explain why they are not groups.

$(\mathbb{Z},+)$ .
$(\mathbb{Z}\setminus\{0\},\times)$ where $\times$ denotes regular multiplication.
$(\mathbb{R}\setminus\{0\},\times)$ where $\times$ denotes regular multiplication.
$(\{0\},\times)$ where $\times$ denotes regular multiplication.
$(\{1\},+)$ where $+$ denotes regular addition.
$(\{1\},\times)$ where $\times$ denotes regular multiplication.

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Problem 2

Show that the trivial group $(\{e\},*)$ , where $e\ast e=e$ , is indeed a group.

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