Groups (8)
Finding this new group in more places
Both the four transformations E,V,P and Q and the four transformations
E,V,R and S work together to give the group D2.
Here's the table for E,V,P and Q.
| |
E |
V |
P |
Q |
| E |
E |
V |
P |
Q |
| V |
V |
E |
Q |
P |
| P |
P |
Q |
E |
V |
| Q |
Q |
P |
V |
E |
You can see that this table has the same structure as D2.
(Remember that C4 shows the structure of the table for E,U,V
and W.)
We can also find the new group D2 inside the group of
the eight symmetry transformations of the square.
The pair of E and V make up a subgroup of
this group. If we rewrite the table so that this subgroup is tucked into
the top left-hand corner, we get the table that I've shown below.
I've treated each block made up of the same letters as a unit,
and used a different colour for each different unit to show how they fit
together.
Here's what the table for the eight symmetry transformations of a
square now looks like.
Combining the 8 symmetry transformations of a
square
The transformation in the vertical column is done first.
| |
E |
V |
U |
W |
P |
Q |
R |
S |
| E |
E |
V |
U |
W |
P |
Q |
R |
S |
| V |
V |
E |
W |
U |
Q |
P |
S |
R |
| U |
U |
W |
V |
E |
S |
R |
P |
Q |
| W |
W |
U |
E |
V |
R |
S |
Q |
P |
| P |
P |
Q |
R |
S |
E |
V |
U |
W |
| Q |
Q |
P |
S |
R |
V |
E |
W |
U |
| R |
R |
S |
Q |
P |
W |
U |
E |
V |
| S |
S |
R |
P |
Q |
U |
W |
V |
E |
You'll see that the coloured blocks show the structure of D2.
E and V form a subgroup and the other pairs of U, W and P, Q and R, S
are what are called cosets.
Is it possible to find infinite numbers of things which obey the rules for
groups? (Of course, we wouldn't even try to show the interactions in a table
if so!)
If we take all the whole numbers and add them in pairs, would this give us
a group?
next