Question

State and prove lagrange's theorm on finite groups?

Answer 1

Answer:

Lagrange theorem is one of the central theorems of abstract algebra. It states that in group theory, for any finite group say G, the order of subgroup H of group G divides the order of G. The order of the group represents the number of elements. This theorem was given by Joseph-Louis Lagrange.

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. Specifically, x and y in G are related if and only if there exists h in H such that x = yh. If we can show that all cosets of H have the same number of elements, then each coset of H has precisely |H| elements. We are then done since the order of H times the number of cosets is equal to the number of elements in G, thereby proving that the order of H divides the order of G.

To show any two left cosets have the same cardinality, it suffices to demonstrate a bijection between them. Suppose aH and bH are two left cosets of H. Then define a map f : aH → bH by setting f(x) = ba−1x.

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