When focusing upon abstract algebra, there are many different areas included within this topic such as groups, rings, modules and vector space. These all are part of the sequence to constructing abstract algebra.
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pure mathematics is mathematics studied and used reasons other than application. So some algebra certainly might be, but not all algebra. Modern or abstract algebra would fit better in that category.
Linear algebra concerns vector spaces whether finite- or infinite-dimensional. Abstract algebra, or modern algebra, includes linear algebra, along with many other kinds of objects, such as groups, rings, fields, lattices, and so on. In part, it was an attempt to put mathematics on a more rigorous footing. Please see the links.
S5 is indecomposable
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