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The Alternating Group a₈ and the General linear Group GL₄(2)

John Murray
Mathematical Proceedings of the Royal Irish Academy
Vol. 99A, No. 2 (Dec., 1999), pp. 123-132
Published by: Royal Irish Academy
Stable URL: http://www.jstor.org/stable/20459753
Page Count: 10
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The Alternating Group a₈ and the General linear Group GL₄(2)
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Abstract

We give an explicit construction for the isomorphism A₈ ≅ GL₄(2). The involutions of cycle type 2³ in the symmetric group S₆, together with the null-set, can be given the structure of an elementary abelian group of order 16, in such a way that S₆ preserves the group operation. This gives an embedding ζ of S₆ into the general linear group GL₄(2). Regarding S₆ as a subgroup of the alternating group A₈, we show that ζ extends to A₈. Coincidence of group orders implies that this extension is an isomorphism.isomorphism.

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