Existence of Maximal Subgroup
Although every proper subgroup of a finite group is contained in a maximal subgroup, there are infinite Abelian groups that contain no maximal subgroups, for example the Prüfer group.
Read more about this topic: Maximal Subgroup
Famous quotes containing the words existence of and/or existence:
“Realism holds that things known may continue to exist unaltered when they are not known, or that things may pass in and out of the cognitive relation without prejudice to their reality, or that the existence of a thing is not correlated with or dependent upon the fact that anybody experiences it, perceives it, conceives it, or is in any way aware of it.”
—William Pepperell Montague (1842–1910)
“It would strike me as ridiculous to want to doubt the existence of Napoleon; but if someone doubted the existence of the earth 150 years ago, perhaps I should be more willing to listen, for now he is doubting our whole system of evidence.”
—Ludwig Wittgenstein (1889–1951)