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Group problem

Source: Romanian District Olympiad 2023 12.2

March 11, 2023

abstract algebragroup theoryalgebra

Problem Statement

Let

(G,\cdot)

be a grup with neutral element

e{}

, and let

H{}

and

K

be proper subgroups of

G

, satisfying

H\cap K=\{e\}

. It is known that

(G\setminus(H\cup K))\cup\{e\}

is closed under the operation of

G

. Prove that

x^2=e

for all the elements

x{}

of

G{}

.

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