Abel's theorem in problems and solutions - School of Mathematics

26 Chapter 1
90. Find the left and right partitions of the group of symmetries of the
square by the following subgroups: a) the subgroup generated by
the central symmetry; b) the subgroup generated by the reflection
with respect to one diagonal.
91. Find the partition of the group of all integers (under addition) 6
by the subgroup of the numbers divisible by 3.
92. Find all groups (up to isomorphism) of order: a) 4; b) 6; c) 8.
1.9 Internal automorphisms
Let us start with an example. Consider the group of symmetries of the
equilateral triangle. If the letters A, B, and C denote the vertices of the
triangle, then each element of the group defines a permutation of the
three letters. For example, the reflection of the triangle with respect to
the altitude drawn from the vertex A to the base BC will be written in the
form To multiply two elements of the group of symmetries
of the triangle it suffices to carry out the corresponding permutations one
after the other. In this way we obtain an isomorphism between the group
of symmetries of the triangle and the group of permutations of letters
A, B, and C.
FIGURE 6
Now we observe that this isomorphism is not uniquely defined: it
depends on which vertex is named A, which B, and which C. The change
of notations of the three vertices of the triangle may be also considered
6 Here we do not mention the type of the partition (left or right) because in commutative
groups the two partitions obviously coincide.