Definition: Let G be a group under a binary operation * and H non empty subset of G. Then H is said to be a subgroup of G if H itself forms a group under the same binary operation * in G. Theorem: A non empty subset H of a ...

Read More »## Dihedral group

Dihedral group: A rotation of the plane lamina of a regular polygon which brings the vertices of the polygon into coincidance with themselves, is called a coincidence rotation or symmetry of the polygon. There are exactly 2n coincidance rotations of a regular polygon of n sides,

Read More »## Show that 0(b) = 1 or 0(b) = 31

Problem: Let a, b be two elements of a group G such that a5 = e and ab a-1 = b2, where e is the identity of G. Show that 0(b) = 1 or 0(b) = 31. Solution: Given that ab a-1 = b2 ————————–(i) (ab a-1)2 = (b2)2 ⟹ ...

Read More »## Show that if every element of the group G execept the identity element is of order 2, then G is abelian.

Solution: Let a, b∈G such that a ≠ e, b ≠ e. Then according to the question, a2 = e, b2 = e. Also ab∈G and So (ab)2 = e Now (ab)2 = e ⟹ (ab) (ab) = e ⟹ a(ab ab) b = a e b ⟹ a2bab2 ...

Read More »## show that G is abelian.

If G is a group in which (ab)1 = a1 b1 for three consecutive integers I for all a, b ∈G, show that G is abelian. Solution: We have (ab)1 = a1 b1 ——————————-(i) (ab)i+1 = ai+1 bi+1 ————————-(ii) (ab)i+2 = ai+2 bi+2 ————————-(iii) From (ii), we have (ab)i+1 = ...

Read More »## The order of any element in G is the same as that of its inverse in G.

Theorem: If G is a group, then (i) for every a ∊ G, 0(a) = 0(a-1). i.e., the order of any element in G is the same as that of its inverse in G. (ii) for any a, x ∊ G. 0(a) = 0(x-1ax) Proof: (i) Let 0(a) = r ...

Read More »## Prove that a^n = e iff m | n( m divides n).

Theorem: Let G be a group with identity e. If a ∊ G and O(a) = m, then for some positive integer n, an = e if and only if m | n (m divides n). Proof: O(a) = m ⇒ m is the least positive integer such that am ...

Read More »## Groups ( Part B)| Modern Abstract Algebra.

The order of an element of group: Let a be a element of a group G. a is said to be of order m, if and only if m is the least positive integer such that am = e (the identity). If there is no positive integer n such that ...

Read More »## Groups| Modern Abstract Algebra

Definition of group: A group is a non empty set G together with a binary operation multiplication (*) such that the following four properties are satisfied (i) Closure Property: a*b∊ G, ∀ a, b ∊G (ii) Associative property: (a * b) * c = a * ( b* c), ∀ ...

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