Let 𝐺 be a group and let 𝑇 < 𝐺. A set Π = {𝐻1, 𝐻2, … , 𝐻𝑛} of proper subgroups of 𝐺 is said to be a 𝑠𝑡𝑟𝑖𝑐𝑡 𝑇 -𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛 of 𝐺, if 𝐺 = ⋃𝑛 𝑖=1 𝐻𝑖 and 𝐻𝑖 ∩ 𝐻𝑗 = 𝑇 for every 1 ≤ 𝑖, 𝑗 ≤ 𝑛. If Π is a strict 𝑇 -partition of 𝐺 and the orders of all components of Π are equal, then we say that 𝐺 has an 𝐸𝑇 -partition. Here we show that: A finite group 𝐺 is nilpotent if and only if every subgroup 𝐻 of 𝐺 has an ES-partition, for some 𝑆 ≤ 𝐻.
