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In literature, I found two differents definition of Primitive Group acting on a set $S$. I report both here:

  1. Let $G$ be a transitive group acting on a set $S$. We say $G$ is primitive if the only blocks for $G$ are the trivial ones.
  2. Let $G$ be a transitive group acting on a set $S$. We say $G$ is primitive if the only $G$-invariant partition are the trivial ones (i.e. $G$ preserves no nontrivial partions of $S$).

Suppose now working with $G = \{ id \}$ ans $S= \{ 1, 2\}$. We easily see that $\{1\}, \{2 \} , S$ are the only blocks for $G$ and moreover $G$ preserves no nontrivial partions of $S$. Then, if we don't consider that 'transitive' in the definitions, we coclude that $G$ is primitive.

Why cut we out this case to be primitive?

More generally, why we refer only to transitive groups when we talk about primitivity? What goes bad if we introduce primitivity without transitivity?

If we suppose $G$ is a permutation group, then it's easy to see that if the only $G$-invariant partition are the trivial ones (or the only blocks for $G$ are the trivial ones) then $G$ is always transitive, except for the case above. It this statement true for general $G$? (using Cayley's Theorem someone could say yes, right?)

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The two definitions you give are equivalent. I guess definitions are chosen to be as convenient as possible to the largest number of people, and the majority of people who work in this area find it most convenient to have primitivity apply only to transitive groups. I suspect that the alternative that you suggest, that we include a single intransitive example, the trivial subgroup of $S_2$, among the primitive groups would cause widespread confusion!

Incidentally, I was initially confused myself by "imprimitive". An imprimitive group is a transitive group that is not primitive, so imprimitive is not the same as not primitive.

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