# # "Is there a group of order 48 in the set of" \ \ 3 xx 3 \ \ "matrices of integers ?" # # "If so, can you exhibit one ? If not, prove its impossibility." #

##### 1 Answer

Yes, for example the group generated by:

#((0, 0, 1),(0, 1, 0),(-1, 0, 0))# ,#((-1, 0, 0),(0, 0, -1),(0,-1,0))# ,#((-1, 0, 0), (0, -1, 0), (0, 0, -1))#

#### Explanation:

Consider the subgroup of

#A = ((0, 0, 1),(0, 1, 0),(-1, 0, 0))#

#B = ((-1, 0, 0),(0, 0, -1),(0,-1,0))#

Notice that

Note that

Between them, these two geometrical operations generate a subgroup of

If we then add a third generator:

#C = ((-1, 0, 0), (0, -1, 0), (0, 0, -1))#

with

we get a subgroup of

**Further reading**

For an in depth analysis of the finite subgroups of