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We have two groups, one of them is automorphisms group of a vector space over GF(2) and another one is the direct product of two automorphism group (they are also over GF(2)). Also, via some computations through GAP, we have the generators of mentioned groups as 6×6 matrices over GF(2). Here is my question: In order to find out whether direct product of automorphism groups yields automorphism group of our vector space, what features of generating matrices must be investigated? If we look at the blocks of matrices, how can we discuss over linear maps and other thing?

Nil
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    I believe this question still makes no sense. Matrices are automorphisms of vector spaces. Do you have some subspaces? Then you are just asking if a subgroup G of GL(6,2) normalizes a subspace V of GF(2)^6. You just check if g*b in V for every generator g of G and basis element b of V. If you want GAP to automate similar questions, you look at the meataxe. – Jack Schmidt Feb 14 '14 at 16:38

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Groups of the form $GL(V)$ for any vector space $V$ over a field $F$ never decompose as direct product of non-trivial groups. So the answer is "no" regardless (unless one of your two automorphism groups is trivial).

Marc van Leeuwen
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  • according to your response, one of the component of direct product must be identity map or identity matrix? – Nil Feb 14 '14 at 13:00
  • One of the factors of the direct product must only contain the identity. In other words the result of the direct product is then just the other factor, and you are just asking whether that other factor coincides with your first automorphism group. – Marc van Leeuwen Feb 14 '14 at 13:03
  • GL(V) actually is sort of often a direct product. The "small" direct factor is central though. Something about n'th roots in the field. GL(2,4) = SL(2,4) x GL(1,4) for example. – Jack Schmidt Feb 14 '14 at 16:26