Question

there exist a subgroup of Z_125 x Z_8 that is isomorphic to Z_5 x Z_5 if yes or not? explain ang show example please!!

Answer #1

Show that S n is isomorphic to a subgroup of A n + 2.

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.

Show that any dihedral group is isomorphic to a subgroup of
GLn(R). (Be as precise as possible,
giving an explicit map between generators r, s of D2n
and certain 2 × 2 matrices.)

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

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

Show that if G is a group, H a subgroup of G with |H| = n, and H
is the only subgroup of G of order n, then H is a normal subgroup
of G.
Hint: Show that aHa-1 is a subgroup of G
and is isomorphic to H for every a ∈ G.

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.

The direct product group R × R has subgroup H = {(5a, a) | a c
R}. Show that group R is isomorphic with group H.

For any integer n>1, prove that Zn[x]/<x> is isomorphic
to Zn. Please explain best way possible and use First Isomorphism
Theorem for rings.

Prove that if A is a subgroup of G and B is a subgroup of H,
then the direct product A × B is a subgroup of G × H.
Show all steps. Note that AXB is nonempty since the identity e
is a part of A X B. Remains only to show that A X B is closed under
multiplication and inverses.

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