Final answer:
Lagrange's Theorem in Group Theory establishes that the order of any subgroup of a finite group divides the group's order. A subgroup's order being a divisor of the group's order is proven by partitioning the group into equal-sized left cosets of the subgroup. The theorem applies to all finite groups and is a proven theorem, not a conjecture or limited to cyclic groups.
Step-by-step explanation:
Lagrange's Theorem in Group Theory states that for any finite group G, the order (the number of elements) of every subgroup H of G divides the order of G. That is, if G is a group with a finite number of elements (order of G), and H is a subgroup of G, then the order of H is a divisor of the order of G.
To prove Lagrange's Theorem, consider the left cosets of H in G. A left coset of H with respect to an element g in G is the set gH = h ∈ H. The left cosets partition the group G such that two cosets are either identical or disjoint and the number of elements in each coset is equal to the order of H. Since G is partitioned into equal-sized cosets of H, the order of G must be a multiple of the order of H, which proves Lagrange's Theorem.
Some clarifications for the given options:
- (a) This is the statement of Lagrange's Theorem which is correct.
- (b) The order of a group is not necessarily the product of the orders of its subgroups; rather, it is the number of distinct left cosets times the order of the subgroup.
- (c) Lagrange's Theorem applies to all finite groups, not only cyclic groups.
- (d) Lagrange's Theorem is a well-established theorem with proof, not a conjecture.