# Group-Theoretic Facts about *G*_{geom} and *G*_{arith}

_{geom}

_{arith}

# Group-Theoretic Facts about *G*_{geom} and *G*_{arith}

_{geom}

_{arith}

This chapter presents proofs of following theorems. Theorem 6.1: Suppose *N* in G_{arith} is geometrically semisimple. Then G_{geom,N} is a normal subgroup of *G _{arith,N}*. Theorem 6.2: Suppose that

*N*in G

_{arith}is arithmetically semisimple and pure of weight zero. If G

_{arith,N}is finite, then

*N*is punctual. Indeed, if every Frobenius conjugacy class FrobE,X in G

_{arith,N}is quasiunipotent, then

*N*is punctual. Theorem 6.4: Suppose that

*N*in G

_{arith}is arithmetically semisimple and pure of weight zero. If G

_{geom}is finite, then

*N*is punctual. Theorem 6.5: Suppose that

*N*in G

_{arith}is arithmetically semisimple and pure of weight zero. Then the group G

_{geom,N}/G⁰

_{geom,N}of connected components of G

_{geom,N}is cyclic of some prime to

*p*order

*n*.

*Keywords:*
number theory, theorems, semisimple, Frobenius conjugacy class

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