Order-cum-power statistics-equivalent not implies 1-isomorphic

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This question was answered on Math Overflow here.

Statement

It is possible to have two groups G and H that are order-cum-power statistics-equivalent (i.e., they have the same order-cum-power statistics) but are not 1-isomorphic.

Proof

Further information: direct product of Q8 and Z4, SmallGroup(32,35)

We construct groups G and H that are both of order 32 = 2^5:

  1. G 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.
  2. H 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 G and H are groups of order 32. 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, G is not 1-isomorphic to H. To see this, note that the three non-identity elements of G have 12, 12, and 4 square roots. On the other hand, the three non-identity elements of H have 20, 4 and 4 square roots. Since the number of square roots of an element must be preserved under a 1-isomorphism, G and H are 1-isomorphic.

References