Jaeger \(et \;al\). [ J. Combin. Theory, Ser B, \(56 (1992) 165-182]\) conjectured that every 3-edge-connected graph is \(Z_5\)-connected. Let \(G\) be a 3-edge-connected simple graph on \(n\) vertices and \(A\) an abelian group with \(|A| \geq 3\). If a graph \(G^*\) is obtained by repeatedly contracting nontrivial \(A\)-connected subgraphs of \(G\) until no such subgraph is left, we say \(G\) can be \(A\)-reduced to \(G^*\). It is proved in this paper that \(G\) is \(A\)-connected with \(|A| \geq 5\) if one of the following holds: (i) \(n \leq 15\); (ii) \(n = 16\) and \(\Delta \geq 4\); or (iii) \(n = 17\) and \(\Delta \geq 5\). As applications, we also show the following results:
(1) For \(|A| \geq 5\) and \(n \geq 17\), if \(|E(G)| \geq \binom{n-15}{2} + 31\), then \(G\) is \(A\)-connected.
(2) For \(|A| \geq 4\) and \(n \geq 13\), if \(|E(G)| \geq \binom{n-11}{2} + 23\), then either \(G\) is \(A\)-connected or \(G\) can be \(A\)-reduced to the Petersen graph.
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