Mathematics Weblog

Similar Groups

Friday 22 April 2005 at 2:57 pm | In Articles | Post Comment

Students learning (finite) group theory often have to prove that 2 groups are isomorphic. They may construct a function from G to H, guided by their Cayley tables, then assume that the function is a homomorphism. Maybe they will check a few cases but don’t think it necessary to prove all |G|^2 equations hold.

They are told that isomorphic groups have the same properties and, in particular, have the same number of elements of the same order. Unfortunately, they assume the converse is true which it isn’t. But the examples they see tend to confirm the converse; they don’t often see counter-examples.

To make things easier let’s say two finite groups G and H are similar if they have the same number of elements of the same order. I suspect this is entirely non-standard terminology 8-).

The counter-example of smallest order, 16, is where $G=C_2 \times C_8$ and H=<a a^2="x^8=1," ax="x^5a"> which are not isomorphic (G is abelian but H isn’t) but both groups have 1 element of order 1, 3 of order 2, 4 of order 4 and 8 of order 8.

Other examples of non-isomorphic similar groups are:

p is an odd prime: $G=C_p \times C_{p^2},\; H=<x x^{p^2}="y^p=1," x^y="x^{1+p}"></x>$ which have elements of order and elements of order
p,q odd primes with $q \equiv 1 \bmod{p}$ . Let x be an element of order p and y, z have order q. Let $P=< x > \cong C_p$ and $Q=< y > \times < z > \cong C_q \times C_q$ . Â G, H are the semi-direct products of Q by P with

G: Â $y^x=y^r,\ z^x=z^r$ where $r^p \equiv 1 \bmod{q},\ r \neq 1$
H: Â $\ y^x=y^r,\ z^x=z^s$ where $r^p \equiv s^p \equiv 1 \bmod{q},\ r,s \neq 1,\ r \not\equiv s \bmod{q}$ Then G, H are non-isomorphic groups of order pq^2 with q^2-1 elements of order and (p-1)q^2 elements of order . The smallest such order is 3^2.7=147

q an odd prime such that $q \equiv 1 \bmod{4}$ . Let x have order 4 and y, z order q. Let $P=< x > \cong C_4$ and $Q=< y > \times < z > \cong C_q \times C_q$ . Â G, H are the semi-direct products of Q by P with

G: Â $y^x=y^r,\ z^x=z^r$
H: Â $y^x=y^r,\ z^x=z^{-r}$ Then G, H are non-isomorphic groups of order 4q^2 with q^2 elements of order 2, 2q^2 elements of order 4 and q^2-1 elements of order . The smallest such order is 4.5^2=100 .