Virtually in the context of Solvable group


Virtually in the context of Solvable group

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⭐ Core Definition: Virtually

In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup such that H has property P.

Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.

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Virtually in the context of Celibacy

Celibacy (from Latin caelibatus) is the state of voluntarily being unmarried, sexually abstinent, or both. It is often in association with the role of a religious official or devotee. In its narrow sense, the term celibacy is applied only to those for whom the unmarried state is the result of a sacred vow, act of renunciation, or religious conviction. In a wider sense, it is commonly understood to only mean abstinence from sexual activity.

Celibacy has existed in one form or another throughout history, in virtually all the major religions of the world, and views on it have varied. The Hindu concept of brahmacharya encourages celibacy during adolescence, to allow one to focus on learning, and in later years, as a way of attaining spiritual liberation. Jainism, on the other hand, preached complete celibacy even for young monks and considered celibacy to be an essential behavior to attain moksha. Buddhism is similar to Jainism in this respect. There were, however, significant cultural differences in the various areas where Buddhism spread, which affected the local attitudes toward celibacy. A somewhat similar situation existed in Japan, where the Shinto tradition also opposed celibacy. In most native African and Native American religious traditions, celibacy has been viewed negatively as well, although there were exceptions like periodic celibacy practiced by some Mesoamerican warriors.

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