Question

Find the group homomorphism from Z_14 to A_4, where A4 is the alternating subgroup of S4. List the elements of A_4

Answer #1

Find the cycle index of the subgroup of S4 consisting of the
permutations of the vertices of a square corresponding to the group
of eight symmetries of the square

(a) Show that H =<(1234)> is a normal subgroup of G=S4
(b) Is the quotient group G/H abelian? Justify?

In the group A4, if Lagrange theorem was applied, how many
cosets would the subgroup generated by (1, 2, 3) have? This
subgroup is denoted by < (1, 2, 3) >. Write down all left
cosets of < (1, 2, 3) >. Is it a group

Find the number of group homomorphism Z_150 to (Z_200 × D_8)

a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2. If
there are elements a, b ∈ G such that ab ∈/ H, then prove that
either a ∈ H or b ∈ H. (b) List the left and right cosets of H =
{(1), (23)} in S3. Are they the same collection?

Can you have a homomorphism from S3 to the group G' with 35
element?

In each part below, a group G and a subgroup H are given.
Determine whether H is normal in G. If it is, list the elements of
the quotient group G/H. (a) G = Z-15 × Z-20 and H = <(10,
17)> (b) G = S-6 and H = A-6 (c) G = S-5 and H = A-4

Let G be a finite group, and suppose that H is normal subgroup
of G.
Show that, for every g ∈ G, the order of gH in G/H must divide
the order of g in G.
What is the order of the coset [4]42 +
〈[6]42〉 in Z42/〈[6]42〉?
Find an example to show that the order of gH in G/H does not
always determine the order of g in G. That is, find an example of a
group G, and...

Where does the initial wall socket alternating current come
from?

Find all elements of order 2 in the dihedral group Dn where n ∈
Z≥3

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