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

Groups 25
Hence left cosets generated by two arbitrary elements either are disjoint
or coincide. In this way we have obtained a partition of all elements
of a group G into disjoint classes. Such a partition is called the left partition
of the group G by the subgroup H.
The number of elements of a subgroup is called the order of the subgroup.
Let be the order of a subgroup H. If and are two different
elements of H then Every left coset thus contains elements.
Hence if is the order of group G and is the number of the left cosets of
the partition of G by H, then and we have proved the following
theorem.
THEOREM 1. (Lagrange’s theorem 5 ) The order of a subgroup H of a
group G divides the order of the group G.
83. Prove that the order of an arbitrary element (see definition in
§1.4) divides the order of the group.
84. Prove that a group whose order is a prime number is cyclic and
that every element of it different from the unit is its generator.
85. Suppose that a group G contains exactly 31 elements. How many
subgroups does it contain?
86. Let be a prime number. Prove that all groups having the same
order are isomorphic to each other.
87. Suppose that divides Obtain a group of order containing
a subgroup isomorphic to a given group G of order
88. Suppose that divides Is it possible that a group of order
does not contain any subgroup of order
One can obtain as well the right cosets and the right partition of
a group G by a subgroup H. If the order of a subgroup H is equal to
then each right coset contains elements and the number of cosets
is equal to the integer where is the order of the group. Hence the
number of right cosets coincides with the number of the left cosets.
89. Find the left and the right partitions of the group of symmetries
of the equilateral triangle by the following subgroups: a) the subgroup
of rotations b) the subgroup generated by the reflection
with respect to one axis.
5 Joseph Louis Lagrange (1736–1813), French mathematician.