= Nontrivial-quotient closure of two-involution groups
Every nontrivial <quotient group> of a <group> generated by two <involutions> is generated by two <involutions>, allowing repeated generators. The two generator images have squares equal to the identity. If one image becomes trivial, the other generates $C_2$; if both become trivial, the quotient is trivial and is excluded. Thus the <two-involution characterization of a dihedral group> is preserved by nontrivial quotients.
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