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On the automorphism group of the integral group ring of the infinite dihedral group

Published online by Cambridge University Press:  14 November 2011

D. A. R. Wallace
D. A. R. Wallace
Affiliation:
University of Strathclyde, Department of Mathematics, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, U.K.
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Synopsis

Let ℤ(G) be the integral group ring of the infinite dihedral group G; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically into M2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t] From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 1 - 10
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

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