Category: 11. Group Theory

A couple of examples of tessellations in geometry are shown below. Basically, whenever you place a polygon together repeatedly without any gaps or overlaps, the resulting figure is a tessellation. A tessellation is also called tiling. Of course, tiles in your house is a real-life example of tessellation. Tessellation with rectangles Tessellation with equilateral triangles…

Let G be a group. A subgroup H of G is said to be a normal subgroup of G if for all h∈ H and x∈ G, x h x-1∈ H If x H x-1 = {x h x-1| h ∈ H} then H is normal in G if and only if xH x-1⊆H, ∀ x∈…

If a non-void subset H of a group G is itself a group under the operation of G, we say H is a subgroup of G. Theorem: – A subset H of a group G is a subgroup of G if: Cyclic Subgroup:- A Subgroup K of a group G is said to be cyclic subgroup…

Let G be a non-void set with a binary operation * that assigns to each ordered pair (a, b) of elements of G an element of G denoted by a * b. We say that G is a group under the binary operation * if the following three properties are satisfied: 1) Associativity: The binary operation…

Let us consider, an algebraic system (A, *), where * is a binary operation on A. Then, the system (A, *) is said to be semi-group if it satisfies the following properties: Example: Consider an algebraic system (A, *), where A = {1, 3, 5, 7, 9….}, the set of positive odd integers and * is…