Basic Examples (5)

Find the multiplicative group of integers modulo 5:

({{"[◼]", "MultiplicativeGroup" }})[5]

See its group multiplication table:

GroupMultiplicationTable[%]//MatrixForm

See the elements of (Subscript[U, 10]):

GroupElements@({{"[◼]", "MultiplicativeGroup" }})[10]

Compute the multiplicative group of various integers:

({{"[◼]", "MultiplicativeGroup" }})/@Range[10]

Multiple cyclic factors emerge for composite inputs:

({{"[◼]", "MultiplicativeGroup" }})[12]

Calculate large groups:

AbsoluteTiming@({{"[◼]", "MultiplicativeGroup" }})[50!]

Calculate its order:

GroupOrder[%[[2]]]

Properties and Relations (7)

Order of MultiplicativeGroup[n] equals EulerPhi[n]:

GroupOrder[({{"[◼]", "MultiplicativeGroup" }})[#]]==EulerPhi[#]&/@{3,4,12,15,101}

For prime p, MultiplicativeGroup[p] evaluates to CyclicGroup[n-1]:

({{"[◼]", "MultiplicativeGroup" }})/@{13,101}

For odd prime powers, MultiplicativeGroup[(p^k)] evaluates to CyclicGroup[(p-1)(p^(k-1))]:

({{"[◼]", "MultiplicativeGroup" }})[(29^10)]

CyclicGroup[(29-1)(29^(10-1))]

For powers of 2, we have:

Table[{(2^k),({{"[◼]", "MultiplicativeGroup" }})[(2^k)]},{k,8}]//TableForm

MultiplicativeGroup[n] where n has multiple distinct prime factors is never cyclic:

({{"[◼]", "MultiplicativeGroup" }})[3 5 7]

MultiplicativeGroup[n] expects n to be a positive integer:

({{"[◼]", "MultiplicativeGroup" }})[0]

MultiplicativeGroup[x] returns unevaluated for arbitrary inputs:

({{"[◼]", "MultiplicativeGroup" }})[x]

%/.x->4

Neat Examples (1)

See the Cayley graph of a large MultiplicativeGroup:

CayleyGraph[({{"[◼]", "MultiplicativeGroup" }})[800]]