Introductory Notions

Part of the book series: UNITEXT ((UNITEXTMAT))

Abstract

Definition 1.1. A group G is a non empty set in which it is defined a binary operation, i.e. a function:

$$ G\times G\to G $$

such that, if ab denotes the image of the pair (a, b),

i)
the operation is associative: (ab)c = a(bc), for all triples of elements a, b, c ∈ G;
ii)
there exists an element e ∈ G such that ea = a = ae, for all a ∈ G. This element is unique: if e′ is also such that e′a = a = ae′, for all a ∈ G, ea = a implies, with a = e′, that ee′ = e′, and a = ae′ implies, with a = e, that ee′ = e. Thus e′ = e;
iii)
for all a ∈ G, there exists b ∈ G such that ab = e = ba.

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1.
From the German Zahl.
2.
Here and in what follows, Ex. means “example” and ex. “exercise”.
3.
Some authors compute the product by first applying τ and then σ (see Remark 1.2).
4.
In practice, the notation GL(n, K) is also used in place of GL(V), and similarly for the other linear groups.
5.
From the German Vierergruppe.
6.
Here and in the sequel (a, b) stands for gcd(a,b).
7.
More precisely, the group of the quaternion units of the skew field of quaternions.
8.
This subgroup is the alternating group (see 2.79).
9.
According to some authors, for example M. Hall jr., these are left cosets.
10.
But the converse is false (Ex. 2.10, 6).
11.
Cf. the paper by Yale P.B.: Automorphisms of the complex numbers. Math. Magazine 39 (1966), 135–141.

Authors

Antonio Machì
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Machì, A. (2012). Introductory Notions. In: Groups. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2421-2_1