If inn(G) denote the inner automophism on G and Dn denote the dihedral group of order 2n.choose the correct option a) inn(D6) contains only two element of order three b) inn(D6) contains only two element of order 6 C) inn(D4) contains only two elements of order 4 D) inn(D6) contains only three elements of order 2?

Question

Answer

Inner Automorphisms and Dihedral Groups

Definition: An automorphism of a group G is an isomorphism from G to itself. The set of all automorphisms of G, denoted Aut(G), forms a group under composition of functions.

Definition: An inner automorphism of a group G is an automorphism of the form φ_g(x) = gxg^(-1), where g is an element of G. The set of all inner automorphisms of G, denoted Inn(G), forms a subgroup of Aut(G).

Dihedral Group: The dihedral group of order 2n, denoted Dn, is the group of symmetries of a regular n-gon. It consists of rotations and reflections of the n-gon.

Statement: We need to determine the number of elements of a specific order in Inn(D6).

Solution:
Inn(D6): To determine the elements of Inn(D6), we need to consider the elements of D6 and their conjugates.

Elements of D6: The elements of D6 can be written as r^i * s^j, where r represents a rotation and s represents a reflection.

Conjugates: Two elements a and b of a group G are conjugates if there exists an element g in G such that b = gag^(-1). Conjugation preserves the order of elements.

Elements of Order 2: The elements of D6 of order 2 are r^3 and s. The inner automorphisms of these elements are given by φ_g(x) = gxg^(-1). Therefore, the elements of Inn(D6) of order 2 are r^3 and s.

Elements of Order 3: The elements of D6 of order 3 are r and r^5. To determine the elements of Inn(D6) of order 3, we need to find the conjugates of these elements. Let's consider the conjugate of r:

r^g = grg^(-1)

By calculating the conjugate of r for different elements of D6, we find that r^g is equal to r^2, r^4, and r^(-1). Similarly, the conjugate of r^5 is r^3, r, and r^(-2). Therefore, the elements of Inn(D6) of order 3 are r^2, r^4, r^(-1), r^3, r, and r^(-2).

Elements of Order 6: The order of an element is the smallest positive integer n such that g^n = e, where e is the identity element. The elements of D6 of order 6 are r and r^5. The inner automorphism of these elements is given by φ_g(x) = gxg^(-1). Therefore, the elements of Inn(D6) of order 6 are r and r^5.

Conclusion: Based on the above analysis, the correct option is (B) inn(D6) contains only two elements of order 6. The other options are not correct as Inn(D6) contains six elements of order 3 and

Answer

Claim: inn(D6) contains only three elements of order 2.

Proof:
To prove this claim, we need to analyze the inner automorphisms of the dihedral group D6.

Definition: An inner automorphism of a group G is defined as conjugation by an element of G.

Conjugation: Conjugation of an element x by an element g in a group G is given by the map c_g(x) = gxg^(-1).

Inner Automorphism: An inner automorphism is a special case of conjugation where the element g and x belong to the same group G.

Dihedral Group of Order 2n: The dihedral group Dn is the group of symmetries of a regular polygon with n sides. It consists of 2n elements - n rotations and n reflections. The rotations form a cyclic group of order n, denoted by Cn, and the reflections form a group of order 2, denoted by Z2. Therefore, Dn = Cn ⋊ Z2, where ⋊ represents the semi-direct product.

Analysis of inn(D6):
To analyze the inner automorphisms of D6, we need to consider the normal subgroup C6 and the quotient group D6/C6.

Normal Subgroup C6: The normal subgroup C6 consists of the rotations of D6. It is a cyclic group of order 6.

Quotient Group D6/C6: The quotient group D6/C6 consists of the cosets of C6 in D6, which are the reflections of D6. It is isomorphic to Z2.

Inner Automorphisms: The inner automorphisms of D6 are given by the conjugation by elements of D6.

Analysis of Order 2 Elements:
For an inner automorphism to have order 2, the conjugating element must have order 2.

Elements of Order 2:
In D6, the elements of order 2 are the reflections, which form a subgroup isomorphic to Z2.

Elements of Order 2 in D6/C6:
In the quotient group D6/C6, the elements of order 2 correspond to the reflections, as they form a subgroup isomorphic to Z2.

Elements of Order 2 in inn(D6):
The elements of order 2 in inn(D6) correspond to the inner automorphisms obtained by conjugating with the reflections of D6.

Conclusion:
Since the reflections in D6 form a subgroup isomorphic to Z2, there are exactly two elements of order 2 in inn(D6). Therefore, the correct option is D) inn(D6) contains only three elements of order 2.

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