Minimal normal subgroup with order not dividing index is characteristic
From Groupprops
Contents
Statement
A minimal normal subgroup of a finite group whose order does not divide its index is characteristic.
Related facts
Facts used
- Automorphisms preserve the property of being normal and hence of being minimal normal.
- Normality is strongly intersection-closed
- Product formula
Proof
Given: A finite group , a minimal normal subgroup
of
. The order of
does not divide the index
of
in
. An automorphism
of
.
To prove: .
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() |
Fact (1) | ![]() ![]() ![]() |
Given-cum-fact direct. | |
2 | ![]() ![]() |
Fact (2) | Step (1) | ![]() ![]() ![]() | |
3 | Either ![]() ![]() |
Steps (1), (2) | By Step (2), ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
4 | If ![]() ![]() ![]() ![]() |
Fact (3) | Step (1) | By Step (1), ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | ![]() ![]() |
Fact (4) | ![]() ![]() |
By Lagrange's theorem, ![]() ![]() ![]() ![]() ![]() | |
6 | ![]() |
Fact (4) | Steps (4), (5) | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | ![]() |
Steps (3) and (6) | Step-combination direct. |