Question

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.

Answer #1

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

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...

Suppose : phi :G -H is a group isomorphism . If N is a normal
subgroup of G then phi(N) is a normal subgroup of H. Prove it is a
subgroup and prove it is normal?

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

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

Let G be a non-trivial finite group, and let H < G be a
proper subgroup. Let X be the set of conjugates of H, that is, X =
{aHa^(−1) : a ∈ G}. Let G act on X by conjugation, i.e., g ·
(aHa^(−1) ) = (ga)H(ga)^(−1) .
Prove that this action of G on X is transitive.
Use the previous result to prove that G is not covered by the
conjugates of H, i.e., G does not equal...

Show that if H is a subgroup of index 2 in a finite group G,
then every left coset of H is also a right coset of H.
*** I have the answer but I am really looking for a thorough
explanation. Thanks!

f H and K are subgroups of a group G, let (H,K) be the subgroup
of G generated by the elements {hkh−1k−1∣h∈H, k∈K}.
Show that :
H◃G if and only if (H,G)<H

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

Let H be a subgroup of the group G. Deﬁne a set B by B = {x ∈ G
| xax−1 ∈ H for all a ∈ H}. Show that H < B.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 4 minutes ago

asked 6 minutes ago

asked 7 minutes ago

asked 13 minutes ago

asked 15 minutes ago

asked 18 minutes ago

asked 21 minutes ago

asked 31 minutes ago

asked 35 minutes ago

asked 35 minutes ago

asked 35 minutes ago