In Group Theory (Abstract Algebra), how do you identify characters/values in a table? If there is any given pattern for it. How do you also find the order of a given group?

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.1: Equations
Problem 50E
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In Group Theory (Abstract Algebra), how do you identify characters/values in a table? If there is any given pattern for it. How do you also find the order of a given group? Example is the picture I've attached :
P
P
P
-
multiplication Table
I
I
C3
63
C₂
T
C3
C²
C₂
C'₂
C"2
C'₂
C₂
for
0
C3.X
C3
2
C3
I
C₂"
C2
C₂
Group D3
C3
2
(3²3²
I
C3
C₂
C₂
C₂
C₂
C2
€₂²
I
C3
I I
iCa
C₂ +
C₂¹
C30
I
C₂
[ix, ix]-
C₂
SSS
C3
(3³²
I
Transcribed Image Text:P P P - multiplication Table I I C3 63 C₂ T C3 C² C₂ C'₂ C"2 C'₂ C₂ for 0 C3.X C3 2 C3 I C₂" C2 C₂ Group D3 C3 2 (3²3² I C3 C₂ C₂ C₂ C₂ C2 €₂² I C3 I I iCa C₂ + C₂¹ C30 I C₂ [ix, ix]- C₂ SSS C3 (3³² I
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