#10.] Let
be fields with
. If
is a subgroup and
is finite, show that
is closed.
Ans:) Given,
.
If
is a subgroup and
is finite.
#10.] Let be fields with . If is a subgroup and is finite, show that is...
Let G be a finite group such that p is a prime and p divides
|G|. Let P be a p-Sylow subgroup of G such that P is cyclic and ?
. Let H be a subgroup of P . Prove
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Define
a prime number,
a finite group,
as a Sylow
-subgroup of
.
Assume there exists
a proper subgroup of
where
, i.e. the normaliser of
in
is a subgroup of
.
Prove that
isn't normal in
.
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Abstract Algebra: Let
. It has been shown already that K is the splitting field over
, and the
following isomorphisms are of onto a subfield
as extensions of the automorphism
, and also the elements of :
;
;
;
.
We also proved previously that is separable over
. Based
on all of those outcomes, find all subgroups of
and their corresponding fixed fields as the intermediate fields
between and
, and
complete the subgroup and subfield diagrams...
Let , ... be independent random variables with mean zero and finite variance. Show that We were unable to transcribe this imageWe were unable to transcribe this image
(10) Let G be a finite group. Prove that if H is a proper subgroup of G, then |H| = |G|/2. (11) Let G be a group. Prove that if Hį and H2 are subgroups of G such that G= H1 U H2, then either H1 = G or H2 = G.
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. [consider the subgroup
of G]
aha а
Define
, a finite
-group, such that
isn't abelian. Let
such that
, where
is abelian.
Prove that there are either
or
such abelian subgroups, and if there are
, then the index of
in
is
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Let G be a finite group and let H be a subgroup of G. Show using double cosets that there is a subset T of G which is simultaneously a left transversal for H and a right transversal for H.
Let
be the orthogonal group of (2 x 2)-matrices over
, and let
be the subset of
.
a) Show that
is a subgroup of
.
b) Show that
is a normal subgroup of
**abstract algebra
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Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...