Question

Q4. Prove that set S of symmetries of a equilateral triangle is a group under composition operation.

Also give reflection symmetries of a square.​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

Set S of symmetries of a equilateral triangle is a group under composition operation.

Step-by-step explanation:

There are 6 symmetric transformation in an equilateral triangle and they form a group.

They are:

1. reflection in the axis through vertex A
2. reflection in the axis through vertex B
3. reflection in the axis through vertex C
4. rotation about the center by 120°
5. rotation about the center by 240°
6. rotation about the center by 360°.  

Rotation about the center by 360° =identity element of the group.

Each reflection is its own inverse.

Let G and H be group:

Isomorphism:

It refers to the one-to-one mapping between G and H groups which helps in group multiplication.  

That is, if g₁ being in G is equivalent to h₁ being in H and

g₁ × g₂ = g₃, then h₁ × h₂ = h₃.

If two groups that are isomorphic ⇒ they are really the same group, represented in two different ways.

Therefore, set S of symmetries of a equilateral triangle is a group under composition operation.

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