Order-cum-power statistics-equivalent not implies 1-isomorphic
This question was answered on Math Overflow here.
It is possible to have two groups and that are order-cum-power statistics-equivalent (i.e., they have the same order-cum-power statistics) but are not 1-isomorphic.
We construct groups and that are both of order :
- is defined as the direct product of Q8 and Z4, i.e., it is the external direct product of the quaternion group and cyclic group of order 4.
- is defined as SmallGroup(32,35). One way of describing this group is as a maximal subgroup of direct product of Q8 and Q8 that is not isomorphic to direct product of Q8 and Z4.
Both and are groups of order . They both have 1 identity element, 3 non-identity elements of order 2 that are squares, and 28 elements of order 4 that are not proper powers. Thus, they have the same order-cum-power statistics.
However, is not 1-isomorphic to . To see this, note that the three non-identity elements of have , , and square roots. On the other hand, the three non-identity elements of have , and square roots. Since the number of square roots of an element must be preserved under a 1-isomorphism, and are 1-isomorphic.