Counterexample bank

#1 · Complex Analysis, Algebra & Topology › Groups

“Converse of Lagrange: d | |G| ⇒ subgroup of order d” — false

Counterexample: A₄ has no subgroup of order 6

#2 · Complex Analysis, Algebra & Topology › Groups

“H ⊴ K and K ⊴ G ⇒ H ⊴ G” — false

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#3 · Complex Analysis, Algebra & Topology › Groups

“Every group of order p² is cyclic” — false

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#4 · Complex Analysis, Algebra & Topology › Groups

“If d divides |G| then G has a subgroup of order d” — false

Counterexample: A₄ has order 12 but no subgroup of order 6

#5 · Complex Analysis, Algebra & Topology › Groups

“o(ab) is finite whenever o(a) and o(b) are” — false

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#6 · Complex Analysis, Algebra & Topology › Groups

“H ⊴ K and K ⊴ G imply H ⊴ G” — false

Counterexample: ⟨(12)(34)⟩ ⊴ V₄ ⊴ A₄

#7 · Complex Analysis, Algebra & Topology › Groups

“If every subgroup of G is normal then G is abelian” — false

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#8 · Complex Analysis, Algebra & Topology › Groups

“Aₙ is simple for every n ≥ 3” — false

Counterexample: A₄ has the normal subgroup V₄ = {e, (12)(34), (13)(24), (14)(23)}

#9 · Complex Analysis, Algebra & Topology › Groups

“A group with trivial centre cannot have order a prime power” — false

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#10 · Complex Analysis, Algebra & Topology › Groups

“G/Z(G) can be cyclic and non-trivial” — false

Counterexample: Impossible — if G/Z is cyclic then G is abelian, so G/Z is trivial

#11 · Complex Analysis, Algebra & Topology › Groups

“A group of order pq (p < q) is always cyclic” — false

Counterexample: S₃ of order 6 = 2·3

#12 · Complex Analysis, Algebra & Topology › Groups

“(ℤ/2^kℤ)* is cyclic for every k” — false

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