## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 35, Number 1 (1994), 1-14.

### Automorphisms of Models of True Arithmetic: Recognizing Some Basic Open Subgroups

Richard Kaye and Henryk Kotlarski

#### Abstract

Let *M* be a countable recursively saturated model of Th($\mathbb{N}$), and let *G*$=$Aut(*M*), considered as a topological group. We examine connections between initial segments of *M* and subgroups of *G*. In particular, for each of the following classes of subgroups *H*$<$*G*, we give characterizations of the class of terms of the topological group structure of *H* as a subgroup of *G*.

(a) $\{ H : H =G_{(K)}$ for some $K\prec_{\rm e} M \}$

(b) $\{ H : H =G_{\{K\}}$ for some $K\prec_{\rm e} M \}$

(c) $\{ H : H =G_{\{M(a)\}}$ for some $ a\in M \}$

(d) $\{ H : H =G_{\{M(a)\}}=G_a$ for some $ a\in M \}$

(Here, *M(a)* denotes the smallest $I\prec_{\rm e}$M containing *a*, $G_{\{A\}}=\{g \in G: A=\{gx: x\in A\} \}$, $G_{(A)}=\{g\in G: \forall a\in A\:ga=a\}$, and $G_a=\{g\in G: ga=a\}$.)

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 35, Number 1 (1994), 1-14.

**Dates**

First available in Project Euclid: 22 December 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1040609291

**Digital Object Identifier**

doi:10.1305/ndjfl/1040609291

**Mathematical Reviews number (MathSciNet)**

MR1271695

**Zentralblatt MATH identifier**

0824.03016

**Subjects**

Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx]

Secondary: 03C15: Denumerable structures 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05] 20B27: Infinite automorphism groups [See also 12F10] 22A05: Structure of general topological groups 54H11: Topological groups [See also 22A05]

#### Citation

Kotlarski, Henryk; Kaye, Richard. Automorphisms of Models of True Arithmetic: Recognizing Some Basic Open Subgroups. Notre Dame J. Formal Logic 35 (1994), no. 1, 1--14. doi:10.1305/ndjfl/1040609291. https://projecteuclid.org/euclid.ndjfl/1040609291