To address the questions regarding characteristic subgroups and their direct powers:

1. Strongest:

If ( H ) is characteristic in ( G ) and ( H \times H ) is characteristic in ( G \times G ), then each ( H^n ) is characteristic in ( G^n ).

Reasoning: For ( H \times H ) to be characteristic in ( G \times G ), ( H ) must be fully invariant in ( G ). Fully invariant sub...

Yes – there is a clean answer that simultaneously addresses your three questions.

Set-up and a standard description of Aut(Gn) For any group G, write Zi = Z(G) for its center. A standard and very useful fact is that Aut(Gn) is generated by:

• permutations of the n coordinates,
• componentwise automorphisms (α1,…,αn) with each αi ∈ Aut(G), and
• “elementary transvections” tij(φ) (i ≠ j) defined by...

This is a fascinating question about the behavior of characteristic subgroups under direct powers. Let me address each of your three questions:

Question 1 (Strongest): Is it true that if H is characteristic in G and H×H is characteristic in G×G, then H^n is characteristic in G^n for all n?

The answer is yes. Here's why:

First, note that if φ: G^n → G^n is an automorphism, then φ can be ...

We write H ≤ G and, for n ≥ 1,

H n = H × … × H (n factors) ≤ G n = G × … × G .

Question (1) If H is characteristic in G and H 2 is characteristic in G 2 , must H n be characteristic in G n for every n ?
Question (2) Does there exist an integer n 0 , depending only on G and H (or only on |G|, or only on the prime p when G is a p-group), such that “ H n characteristic in G n ” has to...

We write all groups multiplicatively.
For a group G and a subgroup H≤G put

  P(n) : “Hn (=H×…×H, n factors) is characteristic in Gn”.

The original example (G = Z8 × Z2, H ={(0,0),(2,1),(4,0),(6,1)}) shows that P(1) does not imply P(2).
The three questions ask whether

(i) P(1)+P(2) ⇒ P(n) for every n ; (ii) there is a fixed n0 so that P(n0) already forces P(n) for all n ;
(iii) there is ...

<think> We are given a solved question that provides a counterexample to the general statement that if H is characteristic in G, then H^n is characteristic in G^n for all n. The counterexample is G = Z_8 × Z_2 and H a specific subgroup of order 4.

Now, the user asks three progressively weaker questions:

1. Strongest: If H is characteristic in G and H × H is characteristic in G × G, then is H^n c...

Characteristic Subgroups and Direct Powers

This question explores a subtle aspect of group theory: when a characteristic subgroup of a group maintains its characteristic property in direct powers.

Understanding the Counterexample

First, let me analyze the given counterexample where and .

is characterist...

This is an excellent set of follow-up questions. The core of the issue lies in the distinction between characteristic subgroups and a stronger condition, being fully invariant. Let's break down the answers, which all hinge on a single key insight.

The Key Insight: The Role of Fully Invariant Subgroups

A subgroup is characteristic if it is invariant under all automorphisms...

Here is the complete picture: one really only needs to check the case n=2, and the property you are demanding is exactly “full‐invariance” of H. In fact one shows:

Lemma. For a subgroup H≤G the following are equivalent.

1. H is fully invariant in G (i.e. φ(H)≤H for every endomorphism φ∶G→G).
2. H×H is characteristic in G×G.
3. For every n≥1 the direct power H^n is characteristic ...