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Block (permutation group theory)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Group_theory/NfowcI5H">group theory</a>, a block system for the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">action</a> of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> <i>G</i> on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> <i>X</i> is a <a href="/facts/Partition_(set_theory)/VeP9g8CA">partition</a> of <i>X</i> that is <i>G</i>-invariant. In terms of the associated <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> on <i>X</i>, <i>G</i>-invariance means that </p> <i>x</i> ~ <i>y</i> implies <i>gx</i> ~ <i>gy</i> <p>for all <i>g</i> ∈ <i>G</i> and all <i>x</i>, <i>y</i> ∈ <i>X</i>. The action of <i>G</i> on <i>X</i> induces a natural action of <i>G</i> on any block system for <i>X</i>. </p><p>The set of <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbits</a> of the <i>G</i>-set <i>X</i> is an example of a block system. The corresponding equivalence relation is the smallest <i>G</i>-invariant equivalence on <i>X</i> such that the induced action on the block system is trivial. </p><p>The partition into <a href="/facts/Singleton_set/LtNVfujT">singleton sets</a> is a block system and if <i>X</i> is non-empty then the partition into one set <i>X</i> itself is a block system as well (if <i>X</i> is a singleton set then these two partitions are identical). A <a href="/facts/Transitive_action/Vh9VtUUw">transitive</a> (and thus non-empty) <i>G</i>-set <i>X</i> is said to be <a href="/facts/Primitive_permutation_group/Qq1nyEwq">primitive</a> if it has no other block systems. For a non-empty <i>G</i>-set <i>X</i> the transitivity requirement in the previous definition is only necessary in the case when |<i>X</i>|=<i>2</i> and the group action is trivial. </p>