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from (xy)^i = x^iy^i show that G is Abelian

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

May 26, 2007

abstract algebragroup theorysuperior algebra

Problem Statement

In the questions below:

G

is a ﬁnite group;

H \leq G

a subgroup of

G; |G : H |

the index of

H

in

G; |X |

the number of elements of

X \subseteq G; Z (G)

the center of

G; G'

the commutator subgroup of

G; N_{G}(H )

the normalizer of

H

in

G; C_{G}(H )

the centralizer of

H

in

G

; and

S_{n}

the

n

-th symmetric group. Suppose

k \geq 2

is an integer such that for all

x, y \in G

and

i \in \{k-1, k, k+1\}

the relation

(xy)^{i}= x^{i}y^{i}

holds. Show that

G

is Abelian.

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