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Miklós Schweitzer 1959- Problem 3

Source:

October 30, 2015

group theoryabstract algebracollege contests

Problem Statement

3.Let

G

be an arbitrary group,

H_1,\dots ,H_n

some (not necessarily distinet) subgroup of

G

and

g_1, \dots , g_n

elements of

G

such that each element of

G

belongs at least to one of the right cosets

H_1 g_1, \dots , H_n g_n

. Show that if, for any

k

, the set-union of the cosets

H_i g_i (i=1, \dots , k-1, k+1, \dots , n)

differs from

G

, then every

H_k (k=1, \dots , n)

is of finite index in

G

. (A. 15)

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