Can a normal subgroup be the group?

A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a characteristic subgroup of a normal subgroup is normal.

How does a quotient group G H define an equivalence relation on G?

In other words, g ·h = gh for all g, h ∈ G, where x denotes the equivalence class of any x ∈ G. Show that ∼ is one of the equivalence relations considered in Problem 1, where S = N is a normal subgroup of G, i.e. G/∼ is nothing but the quotient group G/N.

Is quaternion group normal?

every subgroup of the quaternion group is normal.

Is G always a normal subgroup of itself?

Group is Normal in Itself Let (G,∘) be a group. Then (G,∘) is a normal subgroup of itself.

What is the order of a quotient group?

The order of the quotient group G/H is given by Lagrange Theorem |G/H| = |G|/|H|. and G/H is isomorphic to C2. Example 35. When G = Z, and H = nZ, we cannot use Lagrange since both orders are infinite, still |G/H| = n.

Is a subgroup of order 2 always normal?

Theorem: A subgroup of index 2 is always normal. Proof: Suppose H is a subgroup of G of index 2. Then there are only two cosets of G relative to H . Then G can be decomposed into the cosets H,sH H , s H or H,Hs H , H s , implying H commutes with s .

What is the order of 2 in Z6?

2 has order 2 in Z4, 4 has order 3 in Z12, and 4 has order 3 in Z6. Hence, the order of (2, 4, 4) is [2, 3, 3] = 6. Example. (A product of cyclic groups which is not cyclic) Prove directly that Z2 × Z4 is not cyclic of order 8.

Is Q8 normal?

(c) Since |Q8| = 8, by the Lagrange’s Theorem, any proper subgroup of Q8 has to be of order 2 or 4. Furthermore, any subgroup of order 4 has index 2 in Q8, and hence has to be normal. So it suffices to show that every subgroup of order 2 is normal in Q8.