Quotient-powering-invariance is union-closed

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This article gives the statement, and possibly proof, of a subgroup property (i.e., quotient-powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., union-closed subgroup property)
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Statement

Suppose G is a group and H_i,i \in I is a collection of quotient-powering-invariant subgroups of G. Suppose the union \bigcup_{i \in I} H_i is a subgroup of G (and hence is also the same as the join of subgroups \langle H_i \rangle_{i \in I}. Then, this union of also a quotient-powering-invariant subgroup of G.

Related facts

Facts used

  1. Normality is strongly join-closed

Proof

Note that existence of p^{th} roots is guaranteed (see divisibility is inherited by quotient groups). The part we need to establish is uniqueness. Note also that by Fact (1), the union must be a normal subgroup if it is a subgroup, hence we will assume this in our setup.

Given: G is a group and H_i,i \in I is a collection of quotient-powering-invariant subgroups of G. The union H = \bigcup_{i \in I} H_i is a subgroup of G. G is powered over a prime p. Two elements g_1,g_2 \in G that are in the same coset of H (concretely, g_1g_2^{-1} \in H). x_1,x_2 \in G are respectively the unique solutions to x_1^p = g_1 and x_2^p = g_2.

To prove: x_1 and x_2 are in the same coset of H.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 There exists i \in I such that g_1g_2^{-1} \in H_i. In other words, there exists i \in I such that g_1 and g_2 are in the same coset of H_i. H is the union of H_i, i \in I, g_1g_2^{-1} \in H. Follows directly from given.
2 x_1 and x_2 are in the same coset of H_i. G is p-powered, H_i is quotient-powering-invariant in G. Step (1) Direct from given and Step (1). Note that quotient-powering-invariant in a p-powered group means that the p^{th} roots of elements in the same coset are in the same coset.
3 x_1 and x_2 are in the same coset of H. H is the union of H_i, i \in I. Step (2) Since H_i \le H, being in the same coset of H_i implies being in the same coset of H.