Providing P is prime, we can refer to a [multiplicative] subgroup easily and in keeping with convention.
Let G be the multiplicative group modulo 390001
Let S be the [multiplicative] subgroup generated by 5.
S has 24 elements as shown next
{ 5, 25, 125, 625, 3125, 15625, 78125, 624,
3120, 15600, 78000, -1, 389996, 389976, 389876, 389376,
386876, 374376, 311876, 389377, 386881, 374401, 312001, 1 }
There are two problems potentially as we look at larger / other examples
- The first part G can involve some very large numbers – what happens when you reach 12 digits or more?
- The second part where we talk of subgroups* might be problematic when the modulo is not prime.
*When talking of multiplicative subgroups we are usually doing this in the context of a multiplicative main group.
However when the modulo is composite, there is no group under multiplication from which to subgroup from.
Using sturdy element notation we might refer to the above group instead as
It really is a matter of preference as to which you consider easier.
For this next 24 element example, we are hitting problem 1 (getting large) and problem 2 (composite N) so will not use a G and S way of navigating.
The 24 element multiplicative group
{ 5, 25, 125, 625, 3125,
15625, 78125, 390625, 1953125, 9765625,
48828125, 244140625, 1220703125, 6103515625,
30517578125, 152587890625, 152586328121, 152578515601,
152539453001, 152344140001, 151367575001,
146484750001, 122070625001, 1 }
can only properly be described (in my opinion) using sturdy element notation as shown next.
Where we to try to talk in modulo terms we would be saying modulo 152588281251 for any attempt to describe a main group G.
But we do not have a multiplicative group G?