By Miller G. A.

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**Additional resources for An Overlooked Infinite System of Groups of Order pq2**

**Example text**

If u = (1 + a + a2 )4 − 10ˆ a, then it is not hard to see that conditions (i) and (ii) are satisfied. (d) Let G = a, b | a16 = 1, b2 = 1, ba = a7 b and take H = a2 , b . If u = (1 + a + a2 )8 − 410ˆ a, again conditions (i) and (ii) are satisfied. Cautioning Example. 3 but does NOT give a new grading. Let G be the dihedral group of order 8, that is, G = a, b | a4 = 1, b2 = 1, ba = a3 b . Let H = a2 , b . If u = 1 + (1 − b)a(1 + b) then it is easily checked that u satisfies conditions (i) and (ii).

Math. J. 4 (2004), no. 3, 627–654, 782–783. QA/0301027. W. Ferrer Santos, Cohomology of comodules, Pacific J. Math. 109 (1983), 179– 213. W. Ferrer Santos and A. Rittatore, Actions and invariants of algebraic groups, Series: Pure and Applied Mathematics, 268, Dekker-CRC Press, Florida (2005). J. Fr¨ ohlich and T. Kerler, Quantum groups, quantum categories and quantum field theory, Lecture Notes in Math. 1542, Springer-Verlag, Berlin (1993). R. Hartshorne, Algebraic Geometry, 6th. corrected printing, Springer Verlag, (1993).

Take H = a2 , b . If u = (1 + a + a2 )2 − a ˆ, then uιH (u) = ((1 + a + a2 )2 − a ˆ)((1 − a + a2 )2 − ιH (ˆ a)) = (1 + a2 + a4 )2 − 2(1 + a2 + a4 + a6 ) © 2006 by Taylor & Francis Group, LLC 32 Yu. A. M. Parmenter which is central. To check condition (ii) we need only consider uau−1 − a and ubu−1 − b. The first equals 0 and ubu−1 − b = [(1 + a + a2 )2 − a ˆ]b[(1 + a3 + a6 )2 − a ˆ] − b = [(1 + a + a2 )2 − a ˆ][(1 + a7 + a6 )2 − a ˆ]b − b = [(1 + a + a2 )2 − a ˆ]2 a4 b − b = (−8 − 6a + 6a3 + 9a4 + 6a5 − 6a7 )a4 b − b which is in 2ZG.