Definition of Irreducible Group & Its Relation to GL(n,F)

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An irreducible group is defined as one that cannot be decomposed into a product of two non-trivial subgroups. In the context of GL(n,F), a (semi)group of matrices is considered irreducible if it has no nontrivial invariant subspaces. This definition is relevant to the theorem discussing an irreducible cyclic subgroup of GL(n,F). The discussion highlights the importance of understanding these definitions to apply them correctly in mathematical contexts. Clarifying these terms is essential for accurate interpretation and application in group theory.
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What is the definition of an irreducible group. The context is that I have some theorem talking about "an irreducible cyclic subgroup of GL(n,F)". Is it one that can't be written as a product of two other subgroups in a non-trivial way?
 
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A (semi)group of matrices is said to be irreducible if it has no nontrivial invariant subspaces. At least that's one definition of the word. Does it apply to your case?
 

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