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In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.^[1][2] Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right.^a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004.^aa[›] Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.  

History

The modern concept of an abstract group developed out of several fields of mathematics.^[3][4][5] The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois’ ideas were rejected by his contemporaries, and published only posthumously.^[6][7] More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley‘s On the theory of groups, as depending on the symbolic equation θⁿ = 1 (1854) gives the first abstract definition of a finite group.^[8] Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein‘s 1872 Erlangen program.^[9] After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.^[10] The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss‘ number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker.^[11] In 1847, Ernst Kummer made early attempts to prove Fermat’s Last Theorem by developing groups describing factorization into prime numbers.^[12] The convergence of these various sources into a uniform theory of groups started with Camille Jordan‘s Traité des substitutions et des équations algébriques (1870).^[13] Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an “abstract group”, in the terminology of the time.^[14] As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer‘s modular representation theory and Issai Schur‘s papers.^[15] The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.^[16] Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.^[17] The University of Chicago‘s 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification.^[18] These days, group theory is still a highly active mathematical branch, impacting many other fields.^a[›]

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India (official name: the Republic of India;^[19] Hindi: Bhārat Gaṇarājya) is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast, it shares land borders with Pakistan to the west;^[d] China, Nepal, and Bhutan to the north; and Bangladesh and Myanmar to the east. In the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives; its Andaman and Nicobar Islands share a maritime border with Thailand and Indonesia.

Modern humans arrived on the Indian subcontinent from Africa no later than 55,000 years ago.^[20] Their long occupation, initially in varying forms of isolation as hunter-gatherers, has made the region highly diverse, second only to Africa in human genetic diversity.^[21] Settled life emerged on the subcontinent in the western margins of the Indus river basin 9,000 years ago, evolving gradually into the Indus Valley Civilisation of the third millennium BCE.^[22] By 1200 BCE, an archaic form of Sanskrit, an Indo-European language, had diffused into India from the northwest, unfolding as the language of the Rigveda, and recording the dawning of Hinduism in India.^[23] The Dravidian languages of India were supplanted in the northern regions.^[24] By 400 BCE, stratification and exclusion by caste had emerged within Hinduism,^[25] and Buddhism and Jainism had arisen, proclaiming social orders unlinked to heredity.^[26] Early political consolidations gave rise to the loose-knit Maurya and Gupta Empires based in the Ganges Basin.^[27] Their collective era was suffused with wide-ranging creativity,^[28] but also marked by the declining status of women,^[29] and the incorporation of untouchability into an organized system of belief.^[e][30] In south India, the Middle kingdoms exported Dravidian-languages scripts and religious cultures to the kingdoms of southeast Asia.^[31]

In the early medieval era, Christianity, Islam, Judaism and Zoroastrianism put down roots on India’s southern and western coasts.^[32] Armies from Central Asia intermittently overran India’s plains,^[33] eventually establishing the Delhi sultanate, and drawing northern India into the cosmopolitan networks of medieval Islam.^[34] In the 15th century, the Vijayanagara Empire created a long-lasting composite Hindu culture in south India.^[35] In the Punjab, Sikhism emerged, rejecting institutionalized religion.^[36] The Mughal empire, in 1526, ushered in two centuries of relative peace,^[37] leaving a legacy of luminous architecture.^[f][38] Gradually expanding rule of the British East India Company followed, turning India into a colonial economy, but also consolidating its sovereignty.^[39] British Crown rule began in 1858. The rights promised to Indians were granted slowly,^[40] but technological changes were introduced, and ideas of education, modernity and the public life took root.^[41] A pioneering and influential nationalist movement emerged,^[42] which was noted for nonviolent resistance and led India to its independence in 1947.

India is a secular federal republic governed in a democratic parliamentary system. It is a pluralistic, multilingual and multi-ethnic society. India’s population grew from 361 million in 1951 to 1,211 million in 2011.^[43] During the same time, its nominal per capita income increased from US$64 annually to US$2,041, and its literacy rate from 16.6% to 74%. From being a comparatively destitute country in 1951,^[44] India has become a fast-growing major economy, a hub for information technology services, with an expanding middle class.^[45] It has a space programme which includes several planned or completed lunar missions. Indian movies, music, and spiritual teachings play an increasing role in global culture.^[46] India has substantially reduced its rate of poverty, though at the cost of increasing economic inequality.^[47] India is a nuclear weapons state, which ranks high in military expenditure. It has disputes over Kashmir with its neighbours, Pakistan and China, unresolved since the mid-20th century.^[48] Among the socio-economic challenges India faces are gender inequality, child malnutrition,^[49] and rising levels of air pollution.^[50] India’s land is megadiverse, with four biodiversity hotspots.^[51] Its forest cover comprises 21.4% of its area.^[52] India’s wildlife, which has traditionally been viewed with tolerance in India’s culture,^[53] is supported among these forests, and elsewhere, in protected habitats.