Replacing a subgroup by a normal subgroup

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It often happens that we have a subgroup $H$ of a group $G$ satisfying certain properties, but we want a normal subgroup $N$ of $G$ satisfying the same, or similar, properties. One way of going about this is to try showing that $H$ itself is normal. However, this may be hard to establish, and may in fact be false. There are some other techniques that we can use to replace $H$ by a normal subgroup $N$ of $G$.

For more on techniques to directly prove that a given subgroup is normal, refer to proving normality.

The normal core

Further information: normal core

General approach

The normal core of a subgroup $H$ in a group $G$, denoted $H_G$, is defined as the intersection of all conjugate subgroups of $H$ in $G$. It is also the largest normal subgroup of $G$ contained in $H$, and also equals the kernel of the homomorphism from $G$ to $\operatorname{Sym}(G/H)$ given by the left multiplication action on the left coset space.

Here are some important facts:

• The normal core of $H$ is contained inside $H$. Thus, if we are interested in preserving some property that is subgroup-closed, we can pass to the normal core. For instance, if $H$ is abelian, the normal core of $H$ is abelian. If $H$ is a solvable group, the normal core of $H$ is also solvable.
• The normal core of $H$ is not much smaller than $H$. Specifically, if $H$ has index $n$, the index of the normal core of $H$ divides $n!$. This result is called Poincare's theorem, and it implies that if $H$ has finite index, so does the normal core of $H$. Even in finite groups, this can be used to show that the normal core is a nontrivial normal subgroup, by showing that the index of the normal core is strictly smaller than the order of the group. The latter technique is called the small-index subgroup technique and is used in Sylow theory to show that groups of certain orders cannot be simple.
• The normal core operator also preserves any intersection-closed subgroup property, and for subgroups of finite index, preserves any finite-intersection-closed subgroup property. Also, the normal core of an automorph-conjugate subgroup, and more generally, of a core-characteristic subgroup, is characteristic.

Thus, we can replace a subgroup by its normal core, while preserving subgroup-closed group properties, and not getting too much smaller in the sense that the index of the normal core is bounded in terms of the index of the subgroup.

The normal closure

The normal closure of a subgroup $H$ in a group $G$, denoted $H^G$, is the join of all conjugate subgroups of $H$ in $G$. It is also the smallest normal subgroup of $G$ containing $H$.

The normal closure is a larger subgroup than the original subgroup, but there is no size bound on it in terms of the size of the original subgroup. Thus, it is not possible to guarantee, for instance, that the normal closure of a finite subgroup is finite, or even finitely generated. Also, since the normal subgroup is bigger, it need not satisfy any of the subgroup-closed group properties satisfied by the original subgroup. On the other hand, if a group property is closed under taking joins of groups, then the normal closure of a group satisfying it also satisfies it. For instance, the normal closure of a perfect subgroup is a perfect subgroup.

The normal closure does preserve subgroup properties that are strongly join-closed. Also, it preserves every conjugate-join-closed subgroup property. Other related facts include: The normal closure of an automorph-conjugate subgroup (for instance, a Sylow subgroup) is a characteristic subgroup.

Abelian-to-normal replacement theorems

Replacement theorems are results that allow us to replace one subgroup in a group by another that is similar in some ways and better in others. There are a number of replacement theorems related to groups of prime power order, that guarantee the existence of normal subgroups similar to subgroups we have found. These are discussed below.

Below are given conditions under which, in a group of prime power order $p^n$, we can guarantee that the existence of a subgroup of a certain type of order $p^k$ guarantees the existence of a normal subgroup of the same type and same order.

Satisfaction

Collection of groups of order $p^k$ Condition on prime $p$ Condition on $k$ Proof
Elementary abelian group of order $p^k$ Odd $0 \le k \le 5$ Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
Elementary abelian group of order $p^k$ Odd $k \le (p + 5)/4$ Elementary abelian-to-normal replacement theorem for large primes
Abelian groups of order $p^k$ Odd $0 \le k \le 5$ Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
Abelian groups of order $p^k$, exponent dividing $p^d$ Odd $0 \le d \le k \le 5$ Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
Abelian groups of order $p^k$, exponent dividing $p^d$ Any $0 \le d \le k \le (p + 1)/2$ Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one
Groups of order $p^k$, exponent $p$ -- $k < p$ Mann's replacement theorem for subgroups of prime exponent

Dissatisfaction

Collection of groups of order $p^k$ Condition on prime $p$ Condition on $k$ Proof
Klein four-group $p = 2$ $k = 2$ Elementary abelian-to-normal replacement fails for Klein four-group
Elementary abelian group of order $p^k$ $p \ge 7$ $k \ge (p + 9)/2$ elementary abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Abelian groups of order $p^k$ $p \ge 7$ $k \ge (p + 9)/2$ abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Groups of order $p^p$, exponent $p$ all $p$ $k = p$

Threshold values

This lists threshold values of $k$: the largest value of $k$ for which the collection of $p$-groups of order $p^k$ satisfying the stated condition satisfies a weak normal replacement condition. The nature of all these is such that the weak normal replacement condition is satisfied for all smaller $k$ but for no larger $k$. The between $a$ and $b$ below means that the minimum known value is $a$ and the maximum known value is $b$.

Collection of groups $p = 2$ $p = 3$ $p = 5$ $p = 7$ $p = 11$ $p \ge 11$
Abelian groups of order $p^k$ between 4 and 5 between 5 and 13 between 5 and 6 between 5 and 7 between 6 and 9 between $(p + 1)/2$ and $(p + 7)/2$
Abelian groups of order $p^k$, exponent dividing $p^d, 2 \le d \le k$ between 2 and 5 between 5 and 13 between 5 and 6 between 5 and 7 between 6 and 9 between $(p + 1)/2$ and $(p + 7)/2$
Elementary abelian group of order $p^k$ 1 between 5 and 13 (?) between 5 and 6 (?) between 5 and 7 between 6 and 9 between $(p + 1)/2$ and $(p + 7)/2$
Groups of exponent $p$, order $p^k$ 1 at least 2 at least 4 at least 6

Other replacement theorems

There are some other replacement theorems in both the finite and infinite case. For instance:

Replacement theorem by characteristic subgroup satisfying multilinear commutator identities is a weak analogue for characteristic subgroups to Poincare's theorem, which says that the normal core of a subgroup of index $n$ has index dividing $n!$.