Question

Let G1 and G2 be isomorphic groups.

Prove that if G1 has a subgroup of order n, then G2 has a subgroup of order n

Answer #1

Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2
× G1 (abstract algebra)

Let G1 and G2 be isomorphic groups. Prove each of the
following.
-- If G1 is Abelian, then G2 must be Abelian.
-- If G1 is cyclic, then G2 must be cyclic.

Let G and G′ be two isomorphic groups that have a unique
normal subgroup of a given
order n, H and H′. Show that the quotient groups G/H and G′/H′
are isomorphic.

Let N1 be a normal subgroup of group G1, and N2 be a normal
subgroup of group G2. Use the Fundamental Homomorphism Theorem to
show that
G1/N1 × G2/N2 ≃ (G1 × G2)/(N1 × N2).

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...

Let N be a normal subgroup of G. Show that the order 2 element
in N is in the center of G if N and Z_4 are isomorphic.

Let H be a subgroup of G, and N be the normalizer of H in G and
C be the centralizer of H in G. Prove that C is normal in N and the
group N/C is isomorphic to a subgroup of Aut(H).

Prove that D_4 is also isomorphic to a subgroup H of
S_8. Explicitly list all elements in H in cycle
notations.

Prove that A5 has no subgroup of order 15.

Prove that A5 has no subgroup of order 20.

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