= Higman group
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The Higman group has the <finite group presentation> $Q=\langle a_1,a_2,a_3,a_4\mid a_i a_{i+1}a_i^{-1}=a_{i+1}^2\ (i\bmod4)\rangle$. Its subgroup $\langle a_1,a_3\rangle$ is a rank-two <free group>, and the <normal closure> of each $a_i$ is all of $Q$. It has no nontrivial <finite quotients of a group>: a least prime divisor among the four generator orders in a finite image contradicts the multiplicative-order constraints imposed by conjugation to squares. These properties give finite-presentation constructions embedding arbitrary finitely presented groups into groups with no nontrivial finite quotients.
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