Powering-invariant over quotient-powering-invariant implies powering-invariant
From Groupprops
Statement
Suppose is a group and
are subgroups of
such that
,
is normal in
, and the following conditions hold:
-
is a quotient-powering-invariant subgroup of
.
-
is a powering-invariant subgroup of
.
Then, is a powering-invariant subgroup of
.
Related facts
Proof
Given: is a group and
are subgroups of
such that
,
is normal in
, and the following conditions hold:
-
is a quotient-powering-invariant subgroup of
.
-
is a powering-invariant subgroup of
.
is a prime number such that
is powered over
. An element
.
To prove: There exists such that
.
Proof: Let be the quotient map.
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | There exists ![]() ![]() |
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Given-direct. | ||
2 | ![]() ![]() |
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Given-direct. | ||
3 | ![]() ![]() ![]() ![]() ![]() |
Steps (1), (2) | Applying the homomorphism ![]() ![]() ![]() ![]() ![]() ![]() | ||
4 | ![]() |
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Steps (2), (3) | By Step (2), ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | ![]() |
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Step (4) | Since ![]() ![]() ![]() ![]() |