In: Advanced Math
How do I draw a complete subgroup diagram? The question asks me to give a complete subgroup diagram for Z25x
Let be a group.
The relationships among the various subgroups of the group
can be
illustrated with a subgroup diagram or a
subgroup lattice of the group. This is a
diagram that includes all the subgroups of the group and connects a
subgroup
at one level to a
subgroup
at a higher level with
a sequence of line segments if and only if
is a proper subgroup of
.
Although there are many ways to draw such a diagram, the
connections between the subgroups must be the same. This diagrams
are a very useful way to visuslize a finite group & it's
subgroups.
Now the given group in the question
is .
forms a group under addition modulo n. It is a cyclic group. A
cyclic group of finite order
has one and only one
subgroup of order
for every positive
divisor
of
. Divisors of 25 are 1,
5, 25. Subgroups of
are
(generated by the element
, subgroup
generated by
, &
. Now
. So the subgroup diagram (or lattice) is :-