Counterexample

Is this true?

If d divides |G| then G has a subgroup of order d

No — it is false.

The counterexample

$A_{4}$ has order 12 but no subgroup of order 6

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Hypothesis dropped

A theorem applied without checking one of its preconditions.

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Go deeper· Explained in full in Lagrange, orders and cyclic groups· Asked in CSIR NET June 2025· From Groups › Subgroups, cosets, Lagrange, cyclic groups

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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